Lucas polynomials $L_n$
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Numbers
$n$ 
$L_n(x)$
0:
2
1:
x
2:
x^2 + 2
3:
x^3 + 3*x
4:
x^4 + 4*x^2 + 2
5:
x^5 + 5*x^3 + 5*x
6:
x^6 + 6*x^4 + 9*x^2 + 2
7:
x^7 + 7*x^5 + 14*x^3 + 7*x
8:
x^8 + 8*x^6 + 20*x^4 + 16*x^2 + 2
9:
x^9 + 9*x^7 + 27*x^5 + 30*x^3 + 9*x
10:
x^10 + 10*x^8 + 35*x^6 + 50*x^4 + 25*x^2 + 2
11:
x^11 + 11*x^9 + 44*x^7 + 77*x^5 + 55*x^3 + 11*x
12:
x^12 + 12*x^10 + 54*x^8 + 112*x^6 + 105*x^4 + 36*x^2 + 2
13:
x^13 + 13*x^11 + 65*x^9 + 156*x^7 + 182*x^5 + 91*x^3 + 13*x
14:
x^14 + 14*x^12 + 77*x^10 + 210*x^8 + 294*x^6 + 196*x^4 + 49*x^2 + 2
15:
x^15 + 15*x^13 + 90*x^11 + 275*x^9 + 450*x^7 + 378*x^5 + 140*x^3 + 15*x
16:
x^16 + 16*x^14 + 104*x^12 + 352*x^10 + 660*x^8 + 672*x^6 + 336*x^4 + 64*x^2 + 2
17:
x^17 + 17*x^15 + 119*x^13 + 442*x^11 + 935*x^9 + 1122*x^7 + 714*x^5 + 204*x^3 + 17*x
18:
x^18 + 18*x^16 + 135*x^14 + 546*x^12 + 1287*x^10 + 1782*x^8 + 1386*x^6 + 540*x^4 + 81*x^2 + 2
19:
x^19 + 19*x^17 + 152*x^15 + 665*x^13 + 1729*x^11 + 2717*x^9 + 2508*x^7 + 1254*x^5 + 285*x^3 + 19*x
20:
x^20 + 20*x^18 + 170*x^16 + 800*x^14 + 2275*x^12 + 4004*x^10 + 4290*x^8 + 2640*x^6 + 825*x^4 + 100*x^2 + 2
21:
x^21 + 21*x^19 + 189*x^17 + 952*x^15 + 2940*x^13 + 5733*x^11 + 7007*x^9 + 5148*x^7 + 2079*x^5 + 385*x^3 + 21*x
22:
x^22 + 22*x^20 + 209*x^18 + 1122*x^16 + 3740*x^14 + 8008*x^12 + 11011*x^10 + 9438*x^8 + 4719*x^6 + 1210*x^4 + 121*x^2 + 2
23:
x^23 + 23*x^21 + 230*x^19 + 1311*x^17 + 4692*x^15 + 10948*x^13 + 16744*x^11 + 16445*x^9 + 9867*x^7 + 3289*x^5 + 506*x^3 + 23*x
24:
x^24 + 24*x^22 + 252*x^20 + 1520*x^18 + 5814*x^16 + 14688*x^14 + 24752*x^12 + 27456*x^10 + 19305*x^8 + 8008*x^6 + 1716*x^4 + 144*x^2 + 2
25:
x^25 + 25*x^23 + 275*x^21 + 1750*x^19 + 7125*x^17 + 19380*x^15 + 35700*x^13 + 44200*x^11 + 35750*x^9 + 17875*x^7 + 5005*x^5 + 650*x^3 + 25*x
26:
x^26 + 26*x^24 + 299*x^22 + 2002*x^20 + 8645*x^18 + 25194*x^16 + 50388*x^14 + 68952*x^12 + 63206*x^10 + 37180*x^8 + 13013*x^6 + 2366*x^4 + 169*x^2 + 2
27:
x^27 + 27*x^25 + 324*x^23 + 2277*x^21 + 10395*x^19 + 32319*x^17 + 69768*x^15 + 104652*x^13 + 107406*x^11 + 72930*x^9 + 30888*x^7 + 7371*x^5 + 819*x^3 + 27*x
28:
x^28 + 28*x^26 + 350*x^24 + 2576*x^22 + 12397*x^20 + 40964*x^18 + 94962*x^16 + 155040*x^14 + 176358*x^12 + 136136*x^10 + 68068*x^8 + 20384*x^6 + 3185*x^4 + 196*x^2 + 2
29:
x^29 + 29*x^27 + 377*x^25 + 2900*x^23 + 14674*x^21 + 51359*x^19 + 127281*x^17 + 224808*x^15 + 281010*x^13 + 243542*x^11 + 140998*x^9 + 51272*x^7 + 10556*x^5 + 1015*x^3 + 29*x
30:
x^30 + 30*x^28 + 405*x^26 + 3250*x^24 + 17250*x^22 + 63756*x^20 + 168245*x^18 + 319770*x^16 + 436050*x^14 + 419900*x^12 + 277134*x^10 + 119340*x^8 + 30940*x^6 + 4200*x^4 + 225*x^2 + 2
31:
x^31 + 31*x^29 + 434*x^27 + 3627*x^25 + 20150*x^23 + 78430*x^21 + 219604*x^19 + 447051*x^17 + 660858*x^15 + 700910*x^13 + 520676*x^11 + 260338*x^9 + 82212*x^7 + 14756*x^5 + 1240*x^3 + 31*x
32:
x^32 + 32*x^30 + 464*x^28 + 4032*x^26 + 23400*x^24 + 95680*x^22 + 283360*x^20 + 615296*x^18 + 980628*x^16 + 1136960*x^14 + 940576*x^12 + 537472*x^10 + 201552*x^8 + 45696*x^6 + 5440*x^4 + 256*x^2 + 2
33:
x^33 + 33*x^31 + 495*x^29 + 4466*x^27 + 27027*x^25 + 115830*x^23 + 361790*x^21 + 834900*x^19 + 1427679*x^17 + 1797818*x^15 + 1641486*x^13 + 1058148*x^11 + 461890*x^9 + 127908*x^7 + 20196*x^5 + 1496*x^3 + 33*x
34:
x^34 + 34*x^32 + 527*x^30 + 4930*x^28 + 31059*x^26 + 139230*x^24 + 457470*x^22 + 1118260*x^20 + 2042975*x^18 + 2778446*x^16 + 2778446*x^14 + 1998724*x^12 + 999362*x^10 + 329460*x^8 + 65892*x^6 + 6936*x^4 + 289*x^2 + 2
35:
x^35 + 35*x^33 + 560*x^31 + 5425*x^29 + 35525*x^27 + 166257*x^25 + 573300*x^23 + 1480050*x^21 + 2877875*x^19 + 4206125*x^17 + 4576264*x^15 + 3640210*x^13 + 2057510*x^11 + 791350*x^9 + 193800*x^7 + 27132*x^5 + 1785*x^3 + 35*x
36:
x^36 + 36*x^34 + 594*x^32 + 5952*x^30 + 40455*x^28 + 197316*x^26 + 712530*x^24 + 1937520*x^22 + 3996135*x^20 + 6249100*x^18 + 7354710*x^16 + 6418656*x^14 + 4056234*x^12 + 1790712*x^10 + 523260*x^8 + 93024*x^6 + 8721*x^4 + 324*x^2 + 2
37:
x^37 + 37*x^35 + 629*x^33 + 6512*x^31 + 45880*x^29 + 232841*x^27 + 878787*x^25 + 2510820*x^23 + 5476185*x^21 + 9126975*x^19 + 11560835*x^17 + 10994920*x^15 + 7696444*x^13 + 3848222*x^11 + 1314610*x^9 + 286824*x^7 + 35853*x^5 + 2109*x^3 + 37*x
38:
x^38 + 38*x^36 + 665*x^34 + 7106*x^32 + 51832*x^30 + 273296*x^28 + 1076103*x^26 + 3223350*x^24 + 7413705*x^22 + 13123110*x^20 + 17809935*x^18 + 18349630*x^16 + 14115100*x^14 + 7904456*x^12 + 3105322*x^10 + 810084*x^8 + 128877*x^6 + 10830*x^4 + 361*x^2 + 2
39:
x^39 + 39*x^37 + 702*x^35 + 7735*x^33 + 58344*x^31 + 319176*x^29 + 1308944*x^27 + 4102137*x^25 + 9924525*x^23 + 18599295*x^21 + 26936910*x^19 + 29910465*x^17 + 25110020*x^15 + 15600900*x^13 + 6953544*x^11 + 2124694*x^9 + 415701*x^7 + 46683*x^5 + 2470*x^3 + 39*x
40:
x^40 + 40*x^38 + 740*x^36 + 8400*x^34 + 65450*x^32 + 371008*x^30 + 1582240*x^28 + 5178240*x^26 + 13147875*x^24 + 26013000*x^22 + 40060020*x^20 + 47720400*x^18 + 43459650*x^16 + 29716000*x^14 + 14858000*x^12 + 5230016*x^10 + 1225785*x^8 + 175560*x^6 + 13300*x^4 + 400*x^2 + 2
41:
x^41 + 41*x^39 + 779*x^37 + 9102*x^35 + 73185*x^33 + 429352*x^31 + 1901416*x^29 + 6487184*x^27 + 17250012*x^25 + 35937525*x^23 + 58659315*x^21 + 74657310*x^19 + 73370115*x^17 + 54826020*x^15 + 30458900*x^13 + 12183560*x^11 + 3350479*x^9 + 591261*x^7 + 59983*x^5 + 2870*x^3 + 41*x
42:
x^42 + 42*x^40 + 819*x^38 + 9842*x^36 + 81585*x^34 + 494802*x^32 + 2272424*x^30 + 8069424*x^28 + 22428252*x^26 + 49085400*x^24 + 84672315*x^22 + 114717330*x^20 + 121090515*x^18 + 98285670*x^16 + 60174900*x^14 + 27041560*x^12 + 8580495*x^10 + 1817046*x^8 + 235543*x^6 + 16170*x^4 + 441*x^2 + 2
43:
x^43 + 43*x^41 + 860*x^39 + 10621*x^37 + 90687*x^35 + 567987*x^33 + 2701776*x^31 + 9970840*x^29 + 28915436*x^27 + 66335412*x^25 + 120609840*x^23 + 173376645*x^21 + 195747825*x^19 + 171655785*x^17 + 115000920*x^15 + 57500460*x^13 + 20764055*x^11 + 5167525*x^9 + 826804*x^7 + 76153*x^5 + 3311*x^3 + 43*x
44:
x^44 + 44*x^42 + 902*x^40 + 11440*x^38 + 100529*x^36 + 649572*x^34 + 3196578*x^32 + 12243264*x^30 + 36984860*x^28 + 88763664*x^26 + 169695240*x^24 + 258048960*x^22 + 310465155*x^20 + 292746300*x^18 + 213286590*x^16 + 117675360*x^14 + 47805615*x^12 + 13748020*x^10 + 2643850*x^8 + 311696*x^6 + 19481*x^4 + 484*x^2 + 2
45:
x^45 + 45*x^43 + 945*x^41 + 12300*x^39 + 111150*x^37 + 740259*x^35 + 3764565*x^33 + 14945040*x^31 + 46955700*x^29 + 117679100*x^27 + 236030652*x^25 + 378658800*x^23 + 483841800*x^21 + 488494125*x^19 + 384942375*x^17 + 232676280*x^15 + 105306075*x^13 + 34512075*x^11 + 7811375*x^9 + 1138500*x^7 + 95634*x^5 + 3795*x^3 + 45*x
46:
x^46 + 46*x^44 + 989*x^42 + 13202*x^40 + 122590*x^38 + 840788*x^36 + 4414137*x^34 + 18141618*x^32 + 59198964*x^30 + 154663960*x^28 + 324794316*x^26 + 548354040*x^24 + 741890760*x^22 + 798959280*x^20 + 677688675*x^18 + 445962870*x^16 + 222981435*x^14 + 82317690*x^12 + 21559395*x^10 + 3782350*x^8 + 407330*x^6 + 23276*x^4 + 529*x^2 + 2
47:
x^47 + 47*x^45 + 1034*x^43 + 14147*x^41 + 134890*x^39 + 951938*x^37 + 5154396*x^35 + 21906183*x^33 + 74144004*x^31 + 201619660*x^29 + 442473416*x^27 + 784384692*x^25 + 1120549560*x^23 + 1282801080*x^21 + 1166182800*x^19 + 830905245*x^17 + 455657715*x^15 + 187623765*x^13 + 56071470*x^11 + 11593725*x^9 + 1545830*x^7 + 118910*x^5 + 4324*x^3 + 47*x
48:
x^48 + 48*x^46 + 1080*x^44 + 15136*x^42 + 148092*x^40 + 1074528*x^38 + 5995184*x^36 + 26320320*x^34 + 92285622*x^32 + 260818624*x^30 + 597137376*x^28 + 1109179008*x^26 + 1668903600*x^24 + 2024691840*x^22 + 1965142080*x^20 + 1508593920*x^18 + 901620585*x^16 + 410605200*x^14 + 138389160*x^12 + 33153120*x^10 + 5328180*x^8 + 526240*x^6 + 27600*x^4 + 576*x^2 + 2
49:
x^49 + 49*x^47 + 1127*x^45 + 16170*x^43 + 162239*x^41 + 1209418*x^39 + 6947122*x^37 + 31474716*x^35 + 114191805*x^33 + 334962628*x^31 + 798757036*x^29 + 1551652424*x^27 + 2453288292*x^25 + 3145241400*x^23 + 3247943160*x^21 + 2674776720*x^19 + 1732525830*x^17 + 866262915*x^15 + 326012925*x^13 + 89224590*x^11 + 16921905*x^9 + 2072070*x^7 + 146510*x^5 + 4900*x^3 + 49*x
50:
x^50 + 50*x^48 + 1175*x^46 + 17250*x^44 + 177375*x^42 + 1357510*x^40 + 8021650*x^38 + 37469900*x^36 + 140512125*x^34 + 427248250*x^32 + 1059575660*x^30 + 2148789800*x^28 + 3562467300*x^26 + 4814145000*x^24 + 5272635000*x^22 + 4639918800*x^20 + 3241119750*x^18 + 1767883500*x^16 + 736618125*x^14 + 227613750*x^12 + 50075025*x^10 + 7400250*x^8 + 672750*x^6 + 32500*x^4 + 625*x^2 + 2
51:
x^51 + 51*x^49 + 1224*x^47 + 18377*x^45 + 193545*x^43 + 1519749*x^41 + 9231068*x^39 + 44417022*x^37 + 171986841*x^35 + 541440055*x^33 + 1394538288*x^31 + 2947546836*x^29 + 5114119724*x^27 + 7267433292*x^25 + 8417876400*x^23 + 7887861960*x^21 + 5915896470*x^19 + 3500409330*x^17 + 1602881040*x^15 + 553626675*x^13 + 139299615*x^11 + 24322155*x^9 + 2744820*x^7 + 179010*x^5 + 5525*x^3 + 51*x
52:
x^52 + 52*x^50 + 1274*x^48 + 19552*x^46 + 210795*x^44 + 1697124*x^42 + 10588578*x^40 + 52438672*x^38 + 209456741*x^36 + 681952180*x^34 + 1821786538*x^32 + 4007122496*x^30 + 7262909524*x^28 + 10829900592*x^26 + 13232021400*x^24 + 13160496960*x^22 + 10555815270*x^20 + 6741529080*x^18 + 3370764540*x^16 + 1290244800*x^14 + 366913365*x^12 + 74397180*x^10 + 10145070*x^8 + 851760*x^6 + 38025*x^4 + 676*x^2 + 2
53:
x^53 + 53*x^51 + 1325*x^49 + 20776*x^47 + 229172*x^45 + 1890669*x^43 + 12108327*x^41 + 61669740*x^39 + 253873763*x^37 + 853939021*x^35 + 2363226593*x^33 + 5401660784*x^31 + 10210456360*x^29 + 15944020316*x^27 + 20499454692*x^25 + 21578373360*x^23 + 18443677230*x^21 + 12657425550*x^19 + 6871173870*x^17 + 2893125840*x^15 + 920540040*x^13 + 213696795*x^11 + 34467225*x^9 + 3596580*x^7 + 217035*x^5 + 6201*x^3 + 53*x
54:
x^54 + 54*x^52 + 1377*x^50 + 22050*x^48 + 248724*x^46 + 2101464*x^44 + 13805451*x^42 + 72258318*x^40 + 306312435*x^38 + 1063395762*x^36 + 3045178773*x^34 + 7223447322*x^32 + 14217578856*x^30 + 23206929840*x^28 + 31329355284*x^26 + 34810394760*x^24 + 31604174190*x^22 + 23213240820*x^20 + 13612702950*x^18 + 6263890380*x^16 + 2210784840*x^14 + 580610160*x^12 + 108864405*x^10 + 13741650*x^8 + 1068795*x^6 + 44226*x^4 + 729*x^2 + 2
55:
x^55 + 55*x^53 + 1430*x^51 + 23375*x^49 + 269500*x^47 + 2330636*x^45 + 15696120*x^43 + 84366645*x^41 + 367982175*x^39 + 1317269525*x^37 + 3899117794*x^35 + 9586673915*x^33 + 19619239640*x^31 + 33417386200*x^29 + 47273375600*x^27 + 55309849452*x^25 + 53182547550*x^23 + 41656918050*x^21 + 26270128500*x^19 + 13135064250*x^17 + 5103910680*x^15 + 1501150200*x^13 + 322561200*x^11 + 48208875*x^9 + 4665375*x^7 + 261261*x^5 + 6930*x^3 + 55*x
56:
x^56 + 56*x^54 + 1484*x^52 + 24752*x^50 + 291550*x^48 + 2579360*x^46 + 17797584*x^44 + 98172096*x^42 + 440240493*x^40 + 1623581960*x^38 + 4962513556*x^36 + 12631852688*x^34 + 26842686962*x^32 + 47634965056*x^30 + 70480305440*x^28 + 86639204736*x^26 + 87992942310*x^24 + 73261092240*x^22 + 49483369320*x^20 + 26747767200*x^18 + 11367801060*x^16 + 3711935040*x^14 + 903171360*x^12 + 157073280*x^10 + 18407025*x^8 + 1330056*x^6 + 51156*x^4 + 784*x^2 + 2
57:
x^57 + 57*x^55 + 1539*x^53 + 26182*x^51 + 314925*x^49 + 2848860*x^47 + 20128220*x^45 + 113868216*x^43 + 524607138*x^41 + 1991564135*x^39 + 6279783081*x^37 + 16530970482*x^35 + 36429360877*x^33 + 67254204696*x^31 + 103897691640*x^29 + 133912580336*x^27 + 143302791762*x^25 + 126443639790*x^23 + 91140287370*x^21 + 53017895700*x^19 + 24502865310*x^17 + 8815845720*x^15 + 2404321560*x^13 + 479634480*x^11 + 66615900*x^9 + 5995431*x^7 + 312417*x^5 + 7714*x^3 + 57*x
58:
x^58 + 58*x^56 + 1595*x^54 + 27666*x^52 + 339677*x^50 + 3140410*x^48 + 22707580*x^46 + 131665800*x^44 + 622779234*x^42 + 2431804628*x^40 + 7903365041*x^38 + 21493484038*x^36 + 49061213565*x^34 + 94096891658*x^32 + 151532656696*x^30 + 204392885776*x^28 + 229941996498*x^26 + 214436582100*x^24 + 164401379610*x^22 + 102501265020*x^20 + 51250632510*x^18 + 20183646780*x^16 + 6116256600*x^14 + 1382805840*x^12 + 223689180*x^10 + 24402456*x^8 + 1642473*x^6 + 58870*x^4 + 841*x^2 + 2
59:
x^59 + 59*x^57 + 1652*x^55 + 29205*x^53 + 365859*x^51 + 3455335*x^49 + 25556440*x^47 + 151794020*x^45 + 736647450*x^43 + 2956411766*x^41 + 9894929176*x^39 + 27773267119*x^37 + 65592184047*x^35 + 130526252535*x^33 + 218786861392*x^31 + 308290577416*x^29 + 363854576834*x^27 + 357739373862*x^25 + 290845019400*x^23 + 193641552390*x^21 + 104268528210*x^19 + 44686512090*x^17 + 14932102320*x^15 + 3787127400*x^13 + 703323660*x^11 + 91018356*x^9 + 7637904*x^7 + 371287*x^5 + 8555*x^3 + 59*x
60:
x^60 + 60*x^58 + 1710*x^56 + 30800*x^54 + 393525*x^52 + 3795012*x^50 + 28696850*x^48 + 174501600*x^46 + 868313250*x^44 + 3579191000*x^42 + 12326733804*x^40 + 35676632160*x^38 + 87085668085*x^36 + 179587466100*x^34 + 312883753050*x^32 + 459823234112*x^30 + 568247462610*x^28 + 587681370360*x^26 + 505281601500*x^24 + 358042932000*x^22 + 206769793230*x^20 + 95937144600*x^18 + 35115749100*x^16 + 9903384000*x^14 + 2086129500*x^12 + 314707536*x^10 + 32040360*x^8 + 2013760*x^6 + 67425*x^4 + 900*x^2 + 2
61:
x^61 + 61*x^59 + 1769*x^57 + 32452*x^55 + 422730*x^53 + 4160871*x^51 + 32152185*x^49 + 200058040*x^47 + 1020107270*x^45 + 4315838450*x^43 + 15283145570*x^41 + 45571561336*x^39 + 114858935204*x^37 + 245179650147*x^35 + 443410005585*x^33 + 678610095504*x^31 + 876538040026*x^29 + 951535947194*x^27 + 863020975362*x^25 + 648887951400*x^23 + 400411345620*x^21 + 200205672810*x^19 + 79802261190*x^17 + 24835486320*x^15 + 5873256900*x^13 + 1018031196*x^11 + 123058716*x^9 + 9651664*x^7 + 438712*x^5 + 9455*x^3 + 61*x
62:
x^62 + 62*x^60 + 1829*x^58 + 34162*x^56 + 453530*x^54 + 4554396*x^52 + 35947197*x^50 + 228754890*x^48 + 1194608870*x^46 + 5184151700*x^44 + 18862336570*x^42 + 57898295140*x^40 + 150535567364*x^38 + 332265318232*x^36 + 622997471685*x^34 + 991493848554*x^32 + 1336361274138*x^30 + 1519783409804*x^28 + 1450702345722*x^26 + 1154169552900*x^24 + 758454277620*x^22 + 406975466040*x^20 + 175739405790*x^18 + 59951235420*x^16 + 15776640900*x^14 + 3104160696*x^12 + 437766252*x^10 + 41692024*x^8 + 2452472*x^6 + 76880*x^4 + 961*x^2 + 2
63:
x^63 + 63*x^61 + 1890*x^59 + 35931*x^57 + 485982*x^55 + 4977126*x^53 + 40108068*x^51 + 260907075*x^49 + 1394666910*x^47 + 6204258970*x^45 + 23178175020*x^43 + 73181440710*x^41 + 196107128700*x^39 + 447124253436*x^37 + 868177121832*x^35 + 1434903854139*x^33 + 2014971369642*x^31 + 2396321449830*x^29 + 2402238292916*x^27 + 2017190528262*x^25 + 1407342229020*x^23 + 807386811660*x^21 + 375945078600*x^19 + 139753496610*x^17 + 40612127220*x^15 + 8977417596*x^13 + 1455797448*x^11 + 164750740*x^9 + 12104136*x^7 + 515592*x^5 + 10416*x^3 + 63*x
64:
x^64 + 64*x^62 + 1952*x^60 + 37760*x^58 + 520144*x^56 + 5430656*x^54 + 44662464*x^52 + 296854272*x^50 + 1623421800*x^48 + 7398867840*x^46 + 28362326720*x^44 + 92043777280*x^42 + 254005423840*x^40 + 597659820800*x^38 + 1200442440064*x^36 + 2057901325824*x^34 + 3006465218196*x^32 + 3732682723968*x^30 + 3922021702720*x^28 + 3467892873984*x^26 + 2561511781920*x^24 + 1565841089280*x^22 + 782920544640*x^20 + 315492902400*x^18 + 100563362640*x^16 + 24754058496*x^14 + 4559958144*x^12 + 602516992*x^10 + 53796160*x^8 + 2968064*x^6 + 87296*x^4 + 1024*x^2 + 2
65:
x^65 + 65*x^63 + 2015*x^61 + 39650*x^59 + 556075*x^57 + 5916638*x^55 + 49639590*x^53 + 336962340*x^51 + 1884328875*x^49 + 8793534750*x^47 + 34566585690*x^45 + 115221952300*x^43 + 327186864550*x^41 + 793766949500*x^39 + 1647566693500*x^37 + 2926078447656*x^35 + 4441369072335*x^33 + 5747654093610*x^31 + 6318343152550*x^29 + 5870131166900*x^27 + 4578702310182*x^25 + 2973183318300*x^23 + 1590307356300*x^21 + 691437981000*x^19 + 240316859250*x^17 + 65366185716*x^15 + 13537375740*x^13 + 2058314440*x^11 + 218546900*x^9 + 15072200*x^7 + 602888*x^5 + 11440*x^3 + 65*x
66:
x^66 + 66*x^64 + 2079*x^62 + 41602*x^60 + 593835*x^58 + 6436782*x^56 + 55070246*x^54 + 381624804*x^52 + 2181183147*x^50 + 10416956550*x^48 + 41965453530*x^46 + 143584279020*x^44 + 419230641830*x^42 + 1047772373340*x^40 + 2245226514300*x^38 + 4126520887720*x^36 + 6499270398159*x^34 + 8754119311806*x^32 + 10051025876518*x^30 + 9792152869620*x^28 + 8046595184166*x^26 + 5534695100220*x^24 + 3156148445580*x^22 + 1474358525640*x^20 + 555809761650*x^18 + 165929548356*x^16 + 38291434236*x^14 + 6618272584*x^12 + 821063892*x^10 + 68868360*x^8 + 3570952*x^6 + 98736*x^4 + 1089*x^2 + 2
67:
x^67 + 67*x^65 + 2144*x^63 + 43617*x^61 + 633485*x^59 + 6992857*x^57 + 60986884*x^55 + 431264394*x^53 + 2518145487*x^51 + 12301285425*x^49 + 50758988280*x^47 + 178150864710*x^45 + 534452594130*x^43 + 1374959237890*x^41 + 3038993463800*x^39 + 5774087581220*x^37 + 9425348845815*x^35 + 13195488384141*x^33 + 15798679970128*x^31 + 16110496022170*x^29 + 13916726351066*x^27 + 10113397410402*x^25 + 6129331763880*x^23 + 3064665881940*x^21 + 1247247742650*x^19 + 406246407606*x^17 + 103657619952*x^15 + 20155648324*x^13 + 2879378332*x^11 + 287415260*x^9 + 18643152*x^7 + 701624*x^5 + 12529*x^3 + 67*x
68:
x^68 + 68*x^66 + 2210*x^64 + 45696*x^62 + 675087*x^60 + 7586692*x^58 + 67423666*x^56 + 486334640*x^54 + 2899770291*x^52 + 14482468572*x^50 + 61175944830*x^48 + 220116318240*x^46 + 678036873150*x^44 + 1794189879720*x^42 + 4086765837140*x^40 + 8019314095520*x^38 + 13551869733535*x^36 + 19694758782300*x^34 + 24552799281934*x^32 + 26161521898688*x^30 + 23708879220686*x^28 + 18159992594568*x^26 + 11664026864100*x^24 + 6220814327520*x^22 + 2721606268290*x^20 + 962056169256*x^18 + 269587168308*x^16 + 58447082560*x^14 + 9497650916*x^12 + 1108479152*x^10 + 87511512*x^8 + 4272576*x^6 + 111265*x^4 + 1156*x^2 + 2
69:
x^69 + 69*x^67 + 2277*x^65 + 47840*x^63 + 718704*x^61 + 8220177*x^59 + 74416523*x^57 + 547321524*x^55 + 3331034685*x^53 + 17000614059*x^51 + 73477230255*x^49 + 270875306520*x^47 + 856187737860*x^45 + 2328642473850*x^43 + 5461725075030*x^41 + 11058307559320*x^39 + 19325957314755*x^37 + 29120107628115*x^35 + 37748287666075*x^33 + 41960201868816*x^31 + 39819375242856*x^29 + 32076718945634*x^27 + 21777424274502*x^25 + 12350146091400*x^23 + 5786272150230*x^21 + 2209303911906*x^19 + 675833575914*x^17 + 162104702512*x^15 + 29653299240*x^13 + 3987857484*x^11 + 374926772*x^9 + 22915728*x^7 + 812889*x^5 + 13685*x^3 + 69*x
70:
x^70 + 70*x^68 + 2345*x^66 + 50050*x^64 + 764400*x^62 + 8895264*x^60 + 82003215*x^58 + 614745190*x^56 + 3817369325*x^54 + 19900384350*x^52 + 87959698827*x^50 + 332051251350*x^48 + 1076304056100*x^46 + 3006679347000*x^44 + 7255914954750*x^42 + 15145073396460*x^40 + 27345271410275*x^38 + 42671977361650*x^36 + 57443046448375*x^34 + 66513001150750*x^32 + 65980897141544*x^30 + 55785598166320*x^28 + 39937416869070*x^26 + 24014172955500*x^24 + 12007086477750*x^22 + 4930910180196*x^20 + 1637889745170*x^18 + 431691870820*x^16 + 88100381800*x^14 + 13485508400*x^12 + 1483405924*x^10 + 110427240*x^8 + 5085465*x^6 + 124950*x^4 + 1225*x^2 + 2
71:
x^71 + 71*x^69 + 2414*x^67 + 52327*x^65 + 812240*x^63 + 9613968*x^61 + 90223392*x^59 + 689161713*x^57 + 4364690849*x^55 + 23231419035*x^53 + 104960312886*x^51 + 405528481605*x^49 + 1347179362620*x^47 + 3862867084860*x^45 + 9584557428600*x^43 + 20606798471490*x^41 + 38403578969595*x^39 + 61997934676405*x^37 + 86563154076490*x^35 + 104261288816825*x^33 + 107941099010360*x^31 + 95604973409176*x^29 + 72014135814704*x^27 + 45791597230002*x^25 + 24357232569150*x^23 + 10717182330426*x^21 + 3847193657076*x^19 + 1107525446734*x^17 + 250205084312*x^15 + 43138807640*x^13 + 5471263408*x^11 + 485354012*x^9 + 28001193*x^7 + 937839*x^5 + 14910*x^3 + 71*x
72:
x^72 + 72*x^70 + 2484*x^68 + 54672*x^66 + 862290*x^64 + 10378368*x^62 + 99118656*x^60 + 771164928*x^58 + 4979436039*x^56 + 27048788360*x^54 + 124860697236*x^52 + 493488180432*x^50 + 1679230613970*x^48 + 4939171140960*x^46 + 12591236775600*x^44 + 27862713426240*x^42 + 53548652366055*x^40 + 89343206086680*x^38 + 129235131438140*x^36 + 161704335265200*x^34 + 174454100161110*x^32 + 161585870550720*x^30 + 127799733981024*x^28 + 85729014099072*x^26 + 48371405524650*x^24 + 22724268808176*x^22 + 8778103837272*x^20 + 2745415191904*x^18 + 681896955132*x^16 + 131239189440*x^14 + 18956771808*x^12 + 1968759936*x^10 + 138428433*x^8 + 6023304*x^6 + 139860*x^4 + 1296*x^2 + 2
73:
x^73 + 73*x^71 + 2555*x^69 + 57086*x^67 + 914617*x^65 + 11190608*x^63 + 108732624*x^61 + 861388320*x^59 + 5668597752*x^57 + 31413479209*x^55 + 148092116271*x^53 + 598448493318*x^51 + 2084759095575*x^49 + 6286350503580*x^47 + 16454103860460*x^45 + 37447270854840*x^43 + 74155450837545*x^41 + 127746785056275*x^39 + 191233066114545*x^37 + 248267489341690*x^35 + 278715388977935*x^33 + 269526969561080*x^31 + 223404707390200*x^29 + 157743149913776*x^27 + 94163002754652*x^25 + 47081501377326*x^23 + 19495286167698*x^21 + 6592608848980*x^19 + 1789422401866*x^17 + 381444273752*x^15 + 62095579448*x^13 + 7440023344*x^11 + 623782445*x^9 + 34024497*x^7 + 1077699*x^5 + 16206*x^3 + 73*x
74:
x^74 + 74*x^72 + 2627*x^70 + 59570*x^68 + 969289*x^66 + 12052898*x^64 + 119110992*x^62 + 960506976*x^60 + 6439762680*x^58 + 36392915248*x^56 + 175140904631*x^54 + 723309190554*x^52 + 2578247276007*x^50 + 7965581117550*x^48 + 21393275001420*x^46 + 50038507630440*x^44 + 102018164263785*x^42 + 181295437422330*x^40 + 280576272201225*x^38 + 377502620779830*x^36 + 440419724243135*x^34 + 443981069722190*x^32 + 384990577940920*x^30 + 285542883894800*x^28 + 179892016853724*x^26 + 95452906901976*x^24 + 42219554975874*x^22 + 15370712686252*x^20 + 4534837593770*x^18 + 1063341228884*x^16 + 193334768888*x^14 + 26396795152*x^12 + 2592542381*x^10 + 172452930*x^8 + 7101003*x^6 + 156066*x^4 + 1369*x^2 + 2
75:
x^75 + 75*x^73 + 2700*x^71 + 62125*x^69 + 1026375*x^67 + 12967515*x^65 + 130301600*x^63 + 1069239600*x^61 + 7301151000*x^59 + 42061513000*x^57 + 206554383840*x^55 + 871401306825*x^53 + 3176695769325*x^51 + 10050340213125*x^49 + 27679625505000*x^47 + 66492611490900*x^45 + 139465435118625*x^43 + 255450888259875*x^41 + 408323057257500*x^39 + 568735686894375*x^37 + 688687213584825*x^35 + 722696458700125*x^33 + 654517547502000*x^31 + 508947591285000*x^29 + 337635166767500*x^27 + 189615909656628*x^25 + 89301056353200*x^23 + 34865998853950*x^21 + 11127446442750*x^19 + 2852763630750*x^17 + 574779042640*x^15 + 88492374600*x^13 + 10032565725*x^11 + 796235375*x^9 + 41125500*x^7 + 1233765*x^5 + 17575*x^3 + 75*x
76:
x^76 + 76*x^74 + 2774*x^72 + 64752*x^70 + 1085945*x^68 + 13936804*x^66 + 142354498*x^64 + 1188350592*x^62 + 8261657976*x^60 + 48501275680*x^58 + 242947299088*x^56 + 1046542211456*x^54 + 3900004959879*x^52 + 12628587489132*x^50 + 35645206622550*x^48 + 87885886492320*x^46 + 189503942749065*x^44 + 357469052523660*x^42 + 589618494679830*x^40 + 849311959095600*x^38 + 1066189834364655*x^36 + 1163116182943260*x^34 + 1098498617224190*x^32 + 893938169225920*x^30 + 623178050662300*x^28 + 369507926510352*x^26 + 184753963255176*x^24 + 77085553829824*x^22 + 26498159129002*x^20 + 7387601224520*x^18 + 1638120271524*x^16 + 281827143488*x^14 + 36429360877*x^12 + 3388777756*x^10 + 213578430*x^8 + 8334768*x^6 + 173641*x^4 + 1444*x^2 + 2
77:
x^77 + 77*x^75 + 2849*x^73 + 67452*x^71 + 1148070*x^69 + 14963179*x^67 + 155322013*x^65 + 1318652192*x^63 + 9330897576*x^61 + 55802426680*x^59 + 285008812088*x^57 + 1253096595296*x^55 + 4771406266704*x^53 + 15805283258457*x^51 + 45695546835675*x^49 + 115565511997320*x^47 + 255996554239965*x^45 + 496934487642285*x^43 + 845069382939705*x^41 + 1257635016353100*x^39 + 1634925521259030*x^37 + 1851803396528085*x^35 + 1821195075924315*x^33 + 1548455716727920*x^31 + 1132125641947300*x^29 + 707143093277852*x^27 + 374369872911804*x^25 + 166386610183024*x^23 + 61364157982952*x^21 + 18515047667270*x^19 + 4490883902274*x^17 + 856606186128*x^15 + 124921735477*x^13 + 13421343481*x^11 + 1009813805*x^9 + 49460268*x^7 + 1407406*x^5 + 19019*x^3 + 77*x
78:
x^78 + 78*x^76 + 2925*x^74 + 70226*x^72 + 1212822*x^70 + 16049124*x^68 + 169258817*x^66 + 1461006690*x^64 + 10519248168*x^62 + 64064084656*x^60 + 333510087768*x^58 + 1496043894384*x^56 + 5817948478160*x^54 + 19705288218336*x^52 + 58324134324807*x^50 + 151210718619870*x^48 + 343882440732285*x^46 + 686438430391350*x^44 + 1202538435463365*x^42 + 1847253511032930*x^40 + 2484237480354630*x^38 + 2917993230892740*x^36 + 2984311258867575*x^34 + 2646954333952110*x^32 + 2026063811173220*x^30 + 1330321143940152*x^28 + 743877799422156*x^26 + 351140573438200*x^24 + 138449711812776*x^22 + 45013206796272*x^20 + 11878485126794*x^18 + 2494726457652*x^16 + 406748878965*x^14 + 49850704358*x^12 + 4398591561*x^10 + 263038698*x^8 + 9742174*x^6 + 192660*x^4 + 1521*x^2 + 2
79:
x^79 + 79*x^77 + 3002*x^75 + 73075*x^73 + 1280274*x^71 + 17197194*x^69 + 184221996*x^67 + 1616328703*x^65 + 11837900360*x^63 + 73394982232*x^61 + 389312514448*x^59 + 1781052706472*x^57 + 7071045073456*x^55 + 24476694485040*x^53 + 74129417583264*x^51 + 196906265455545*x^49 + 459447952729605*x^47 + 942434984631315*x^45 + 1699472923105650*x^43 + 2692322893972635*x^41 + 3741872496707730*x^39 + 4552918752151770*x^37 + 4836114655395660*x^35 + 4468149409876425*x^33 + 3574519527901140*x^31 + 2462446785887452*x^29 + 1451020892700008*x^27 + 725510446350004*x^25 + 304836321995800*x^23 + 106377364779224*x^21 + 30393532794064*x^19 + 6985610359926*x^17 + 1263355065093*x^15 + 174772439835*x^13 + 17819935042*x^11 + 1272852503*x^9 + 59202442*x^7 + 1600066*x^5 + 20540*x^3 + 79*x
80:
x^80 + 80*x^78 + 3080*x^76 + 76000*x^74 + 1350500*x^72 + 18410016*x^70 + 200271120*x^68 + 1785587520*x^66 + 13298907050*x^64 + 83914230400*x^62 + 453376599104*x^60 + 2114562794240*x^58 + 8567088967840*x^56 + 30294642963200*x^54 + 93834705801600*x^52 + 255230399780352*x^50 + 610658671349475*x^48 + 1286317425363600*x^46 + 2385911353497000*x^44 + 3894861329436000*x^42 + 5589126007740660*x^40 + 7037156232506400*x^38 + 7754107886288400*x^36 + 7452460668744000*x^34 + 6221473861853250*x^32 + 4488510597060672*x^30 + 2781342036640160*x^28 + 1469388245772160*x^26 + 655976895434000*x^24 + 244827076592000*x^22 + 75406739590336*x^20 + 18864095486720*x^18 + 3758081522745*x^16 + 581521318800*x^14 + 67670639400*x^12 + 5671444064*x^10 + 322241140*x^8 + 11342240*x^6 + 213200*x^4 + 1600*x^2 + 2
81:
x^81 + 81*x^79 + 3159*x^77 + 79002*x^75 + 1423575*x^73 + 19690290*x^71 + 217468314*x^69 + 1969809516*x^67 + 14915235753*x^65 + 95752130760*x^63 + 526771581336*x^61 + 2503875308688*x^59 + 10348141674312*x^57 + 37365688036656*x^55 + 118311400286640*x^53 + 329359817363616*x^51 + 807564936805020*x^49 + 1745765378093205*x^47 + 3328346338128315*x^45 + 5594334252541650*x^43 + 8281448901713295*x^41 + 10779028729214130*x^39 + 12307026638440170*x^37 + 12288575324139660*x^35 + 10689623271729675*x^33 + 8063030124961812*x^31 + 5243788822527612*x^29 + 2920409138472168*x^27 + 1381487341784004*x^25 + 549663398587800*x^23 + 181784104369560*x^21 + 49257628280784*x^19 + 10743691882671*x^17 + 1844876383893*x^15 + 242443079235*x^13 + 23491379106*x^11 + 1595093643*x^9 + 70544682*x^7 + 1813266*x^5 + 22140*x^3 + 81*x
82:
x^82 + 82*x^80 + 3239*x^78 + 82082*x^76 + 1499575*x^74 + 21040790*x^72 + 235878330*x^70 + 2170080636*x^68 + 16700823273*x^66 + 109051037810*x^64 + 610685811736*x^62 + 2957251907792*x^60 + 12462704468552*x^58 + 45932777004496*x^56 + 148606043249840*x^54 + 423194523165216*x^52 + 1062795336585372*x^50 + 2356424049442680*x^48 + 4614663763491915*x^46 + 7980245606038650*x^44 + 12176310231149295*x^42 + 16368154736954790*x^40 + 19344182870946570*x^38 + 20042683210428060*x^36 + 18142083940473675*x^34 + 14284503986815062*x^32 + 9732299419588284*x^30 + 5701751175112328*x^28 + 2850875587556164*x^26 + 1205640294021800*x^24 + 426611180961560*x^22 + 124664367871120*x^20 + 29607787369391*x^18 + 5602957906638*x^16 + 823964398035*x^14 + 91162018506*x^12 + 7266537707*x^10 + 392785822*x^8 + 13155506*x^6 + 235340*x^4 + 1681*x^2 + 2
83:
x^83 + 83*x^81 + 3320*x^79 + 85241*x^77 + 1578577*x^75 + 22464365*x^73 + 255568620*x^71 + 2387548950*x^69 + 18670632789*x^67 + 123966273563*x^65 + 706437942496*x^63 + 3484023489128*x^61 + 14966579777240*x^59 + 56280918678808*x^57 + 185971731286496*x^55 + 541505923451856*x^53 + 1392155153948988*x^51 + 3163988986247700*x^49 + 6360429141585120*x^47 + 11308591944166965*x^45 + 17770644483690945*x^43 + 24649603638668085*x^41 + 30123211600160700*x^39 + 32349709848868230*x^37 + 30430659264613335*x^35 + 24974127258544737*x^33 + 17795329544550096*x^31 + 10945539997639940*x^29 + 5771284726028332*x^27 + 2587127635805804*x^25 + 976274579549360*x^23 + 306448472240680*x^21 + 78865415650175*x^19 + 16346649789309*x^17 + 2668840781928*x^15 + 333605097741*x^13 + 30757916813*x^11 + 1987879465*x^9 + 83700188*x^7 + 2048606*x^5 + 23821*x^3 + 83*x
84:
x^84 + 84*x^82 + 3402*x^80 + 88480*x^78 + 1660659*x^76 + 23963940*x^74 + 276609410*x^72 + 2623427280*x^70 + 20840713425*x^68 + 140667096836*x^66 + 815488980306*x^64 + 4094709300864*x^62 + 17923831685032*x^60 + 68743623147360*x^58 + 231904508290992*x^56 + 690111966701696*x^54 + 1815349677114204*x^52 + 4226784322833072*x^50 + 8716853191027800*x^48 + 15923255707658880*x^46 + 25750890089729595*x^44 + 36825913869817380*x^42 + 46491366337115490*x^40 + 51693892719814800*x^38 + 50473342475041395*x^36 + 43116211199018412*x^34 + 32079833531365158*x^32 + 20677839417228224*x^30 + 11473035901140660*x^28 + 5438003223361968*x^26 + 2181914873571160*x^24 + 733059653202240*x^22 + 203529783521295*x^20 + 45954437158700*x^18 + 8271798688566*x^16 + 1157569495776*x^14 + 121919935319*x^12 + 9254417172*x^10 + 476486010*x^8 + 15204112*x^6 + 259161*x^4 + 1764*x^2 + 2
85:
x^85 + 85*x^83 + 3485*x^81 + 91800*x^79 + 1745900*x^77 + 25542517*x^75 + 299073775*x^73 + 2878995900*x^71 + 23228262375*x^69 + 159337729625*x^67 + 939455253869*x^65 + 4801147243360*x^63 + 21407855174160*x^61 + 83710202924600*x^59 + 288185426969800*x^57 + 876083697988192*x^55 + 2356855600566060*x^53 + 5618939476782060*x^51 + 11880842177275500*x^49 + 22283684849244000*x^47 + 37059482033896560*x^45 + 54596558353508325*x^43 + 71140969975783575*x^41 + 81817104319975500*x^39 + 82823052323909625*x^37 + 73546870463631747*x^35 + 57053960789909895*x^33 + 38473168961778320*x^31 + 22418575898780600*x^29 + 11209287949390300*x^27 + 4769042509376964*x^25 + 1709334232751600*x^23 + 509978255761975*x^21 + 124819852808875*x^19 + 24618448477875*x^17 + 3826410277704*x^15 + 455525033060*x^13 + 40012333985*x^11 + 2464365475*x^9 + 98904300*x^7 + 2307767*x^5 + 25585*x^3 + 85*x
86:
x^86 + 86*x^84 + 3569*x^82 + 95202*x^80 + 1834380*x^78 + 27203176*x^76 + 323037715*x^74 + 3155605310*x^72 + 25851689655*x^70 + 180178443050*x^68 + 1080122350705*x^66 + 5616636223666*x^64 + 25502564475024*x^62 + 101634034609632*x^60 + 356929050117160*x^58 + 1107988206279184*x^56 + 3046967567267756*x^54 + 7434289153896264*x^52 + 16107626500108572*x^50 + 31000538040271800*x^48 + 52982737741555440*x^46 + 80347448443237920*x^44 + 107966883845600955*x^42 + 128308470657090990*x^40 + 134516945043724425*x^38 + 124020212938673142*x^36 + 100170171988928307*x^34 + 70553002493143478*x^32 + 43096415316008824*x^30 + 22682323850530960*x^28 + 10207045732738932*x^26 + 3891249106322760*x^24 + 1243037908964215*x^22 + 328349636330170*x^20 + 70572885636575*x^18 + 12098208966270*x^16 + 1613094528836*x^14 + 161932269304*x^12 + 11718782647*x^10 + 575390310*x^8 + 17511879*x^6 + 284746*x^4 + 1849*x^2 + 2
87:
x^87 + 87*x^85 + 3654*x^83 + 98687*x^81 + 1926180*x^79 + 28949076*x^77 + 348580232*x^75 + 3454679085*x^73 + 28730685555*x^71 + 203406705425*x^69 + 1239460080330*x^67 + 6556091477535*x^65 + 30303711718384*x^63 + 123041889783792*x^61 + 440639253041760*x^59 + 1396173633248984*x^57 + 3923051265255948*x^55 + 9791144754462324*x^53 + 21726565976890632*x^51 + 42881380217547300*x^49 + 75266422590799440*x^47 + 117406930477134480*x^45 + 162563442199109280*x^43 + 199449440632874565*x^41 + 216334049363699925*x^39 + 206843265262582767*x^37 + 173717042452560054*x^35 + 127606963283053373*x^33 + 81569584277787144*x^31 + 45100899749311560*x^29 + 21416333682129232*x^27 + 8660291615699724*x^25 + 2952372141715815*x^23 + 838327892092145*x^21 + 195392738445450*x^19 + 36716657444145*x^17 + 5439504806540*x^15 + 617457302364*x^13 + 51731116632*x^11 + 3039755785*x^9 + 116416179*x^7 + 2592513*x^5 + 27434*x^3 + 87*x
88:
x^88 + 88*x^86 + 3740*x^84 + 102256*x^82 + 2021382*x^80 + 30783456*x^78 + 375783408*x^76 + 3777716800*x^74 + 31886290865*x^72 + 229258395080*x^70 + 1419638523380*x^68 + 7636213828240*x^66 + 35920347942050*x^64 + 148544454258816*x^62 + 542273287651392*x^60 + 1753102683366144*x^58 + 5031039471535132*x^56 + 12838112321730080*x^54 + 29160855130786896*x^52 + 58989006717655872*x^50 + 106266960631071240*x^48 + 170389668218689920*x^46 + 242910890642347200*x^44 + 307416324478475520*x^42 + 344642520020790915*x^40 + 341360210306307192*x^38 + 297737255391233196*x^36 + 227777135271981680*x^34 + 152122586770930622*x^32 + 88197315065320384*x^30 + 44098657532660192*x^28 + 18867337348438656*x^26 + 6843621248038575*x^24 + 2081365801056360*x^22 + 523742374775620*x^20 + 107289543080720*x^18 + 17537713772810*x^16 + 2230551831200*x^14 + 213663385936*x^12 + 14758538432*x^10 + 691806489*x^8 + 20104392*x^6 + 312180*x^4 + 1936*x^2 + 2
89:
x^89 + 89*x^87 + 3827*x^85 + 105910*x^83 + 2120069*x^81 + 32709636*x^79 + 404732484*x^77 + 4126297032*x^75 + 35340969950*x^73 + 257989080635*x^71 + 1623045228805*x^69 + 8875673908570*x^67 + 42476439419585*x^65 + 178848165977200*x^63 + 665315177435184*x^61 + 2193741936407904*x^59 + 6427213104784116*x^57 + 16761163586986028*x^55 + 38951999885249220*x^53 + 80715572694546504*x^51 + 149148340848618540*x^49 + 245656090809489360*x^47 + 360317821119481680*x^45 + 469979766677584800*x^43 + 544091960653665480*x^41 + 557694259670007117*x^39 + 504580520653815963*x^37 + 401494177724541734*x^35 + 279729550053983995*x^33 + 169766899343107528*x^31 + 89199557281971752*x^29 + 40283671030567888*x^27 + 15503912863738299*x^25 + 5033737942772175*x^23 + 1362070266867765*x^21 + 302682281526170*x^19 + 54254371216955*x^17 + 7670056637740*x^15 + 831120688300*x^13 + 66489655064*x^11 + 3731562274*x^9 + 136520571*x^7 + 2904693*x^5 + 29370*x^3 + 89*x
90:
x^90 + 90*x^88 + 3915*x^86 + 109650*x^84 + 2222325*x^82 + 34731018*x^80 + 435515940*x^78 + 4502080440*x^76 + 39118686750*x^74 + 289875371500*x^72 + 1852303623885*x^70 + 10295312431950*x^68 + 50112653247825*x^66 + 214768513919250*x^64 + 813859631694000*x^62 + 2736015224059296*x^60 + 8180315788150260*x^58 + 21792203058521160*x^56 + 51790112206979300*x^54 + 109876427825333400*x^52 + 208137347566274412*x^50 + 351923051440560600*x^48 + 530707489338171600*x^46 + 712890657319932000*x^44 + 851508285132141000*x^42 + 902336779690798032*x^40 + 845940730960123155*x^38 + 699231433115774930*x^36 + 507506685325965675*x^34 + 321889486114038150*x^32 + 177396872347292136*x^30 + 84382328563228080*x^28 + 34371250212176955*x^26 + 11877359190810750*x^24 + 3443436067924125*x^22 + 826424656301790*x^20 + 161543914297675*x^18 + 25207770410550*x^16 + 3061672519500*x^14 + 280153041000*x^12 + 18490100706*x^10 + 828327060*x^8 + 23009085*x^6 + 341550*x^4 + 2025*x^2 + 2
91:
x^91 + 91*x^89 + 4004*x^87 + 113477*x^85 + 2328235*x^83 + 36851087*x^81 + 468225576*x^79 + 4906812924*x^77 + 43244983782*x^75 + 325216341450*x^73 + 2110292704520*x^71 + 11918357660755*x^69 + 58988327156395*x^67 + 257244953338835*x^65 + 992707797671200*x^63 + 3401330401494480*x^61 + 10374057724558164*x^59 + 28219416163305276*x^57 + 68551275793965328*x^55 + 148828427710582620*x^53 + 288852920260820916*x^51 + 501071392289179140*x^49 + 776363580147660960*x^47 + 1073208478439413680*x^45 + 1321488051809725800*x^43 + 1446428740344463512*x^41 + 1403634990630130272*x^39 + 1203811953769590893*x^37 + 909000863050507409*x^35 + 601619036168022145*x^33 + 347163771690399664*x^31 + 173581885845199832*x^29 + 74654921242744843*x^27 + 27381272054549049*x^25 + 8477174010696300*x^23 + 2188494923169555*x^21 + 464226195823845*x^19 + 79462141627505*x^17 + 10731729157240*x^15 + 1111273729300*x^13 + 84979755770*x^11 + 4559889334*x^9 + 159529656*x^7 + 3246243*x^5 + 31395*x^3 + 91*x
92:
x^92 + 92*x^90 + 4094*x^88 + 117392*x^86 + 2437885*x^84 + 39073412*x^82 + 502956594*x^80 + 5342328864*x^78 + 47747064222*x^76 + 364335028200*x^74 + 2400168076020*x^72 + 13770661284640*x^70 + 69283639588345*x^68 + 307357606586660*x^66 + 1207476311590450*x^64 + 4215190033188480*x^62 + 13110072948617460*x^60 + 36399731951455536*x^58 + 90343478852486488*x^56 + 200618539917561920*x^54 + 398729348086154316*x^52 + 709208739855453552*x^50 + 1128286631588221560*x^48 + 1603915967777585280*x^46 + 2034378709129657800*x^44 + 2297937025476604512*x^42 + 2305971770320928304*x^40 + 2049752684729714048*x^38 + 1608232296166282339*x^36 + 1109125721493987820*x^34 + 669053257804437814*x^32 + 350978758192491968*x^30 + 159037249805972923*x^28 + 61752522266726004*x^26 + 20354533201507050*x^24 + 5631930991093680*x^22 + 1290650852125635*x^20 + 241006055925180*x^18 + 35939499567790*x^16 + 4172946248800*x^14 + 365132796770*x^12 + 23049990040*x^10 + 987856716*x^8 + 26255328*x^6 + 372945*x^4 + 2116*x^2 + 2
93:
x^93 + 93*x^91 + 4185*x^89 + 121396*x^87 + 2551362*x^85 + 41401647*x^83 + 539807681*x^81 + 5810554440*x^79 + 52653877146*x^77 + 407580011982*x^75 + 2725384417470*x^73 + 15880953989160*x^71 + 81201997249100*x^69 + 366345933743055*x^67 + 1464721264929285*x^65 + 5207897830859680*x^63 + 16511403350111940*x^61 + 46773789676013700*x^59 + 118562895015791764*x^57 + 269169815711527248*x^55 + 547557775796736936*x^53 + 998061660116274468*x^51 + 1629358023877400700*x^49 + 2380279547925246240*x^47 + 3107587187569071480*x^45 + 3619425077286330312*x^43 + 3752400510665391816*x^41 + 3453387675359844320*x^39 + 2812044249935873232*x^37 + 2018126584544495229*x^35 + 1270672293972459959*x^33 + 698142529882891632*x^31 + 332619135651172755*x^29 + 136407443509470847*x^27 + 47735805256056099*x^25 + 14109105001789980*x^23 + 3479145775295190*x^21 + 705232251749025*x^19 + 115401641195295*x^17 + 14904675406040*x^15 + 1476406526070*x^13 + 108029745810*x^11 + 5547746050*x^9 + 185784984*x^7 + 3619188*x^5 + 33511*x^3 + 93*x
94:
x^94 + 94*x^92 + 4277*x^90 + 125490*x^88 + 2668754*x^86 + 43839532*x^84 + 578881093*x^82 + 6313511034*x^80 + 57996206010*x^78 + 455327076204*x^76 + 3089719445670*x^74 + 18281122065180*x^72 + 94972658533740*x^70 + 435629573331400*x^68 + 1772078871515945*x^66 + 6415374142450130*x^64 + 20726593383300420*x^62 + 59883862624631160*x^60 + 154962626967247300*x^58 + 359513294564013736*x^56 + 748176315714298856*x^54 + 1396791008202428784*x^52 + 2338566763732854252*x^50 + 3508566179513467800*x^48 + 4711503155346656760*x^46 + 5653803786415988112*x^44 + 6050337536141996328*x^42 + 5759359445680772624*x^40 + 4861796934665587280*x^38 + 3626358880710777568*x^36 + 2379798015466447779*x^34 + 1367195787687329446*x^32 + 683597893843664723*x^30 + 295444693315443770*x^28 + 109488327522782103*x^26 + 34463638203297030*x^24 + 9111076766388870*x^22 + 1995883103874660*x^20 + 356407697120475*x^18 + 50844174973830*x^16 + 5649352774870*x^14 + 473162542580*x^12 + 28597736090*x^10 + 1173641700*x^8 + 29874516*x^6 + 406456*x^4 + 2209*x^2 + 2
95:
x^95 + 95*x^93 + 4370*x^91 + 129675*x^89 + 2790150*x^87 + 46390894*x^85 + 620282740*x^83 + 6853318715*x^81 + 63806760450*x^79 + 507980953350*x^77 + 3497299457652*x^75 + 21006506482650*x^73 + 110853612522900*x^71 + 516831570580500*x^69 + 2138424805259000*x^67 + 7880095407379415*x^65 + 25934491214160100*x^63 + 76395265974743100*x^61 + 201736416643261000*x^59 + 478076189579805500*x^57 + 1017346131425826104*x^55 + 1944348783999165720*x^53 + 3336628423849128720*x^51 + 5137924203390868500*x^49 + 7091782703271903000*x^47 + 8761390973985059592*x^45 + 9669762613428326640*x^43 + 9511759956346164440*x^41 + 8315184610025431600*x^39 + 6438403130646650800*x^37 + 4397924600010943008*x^35 + 2637868081659789405*x^33 + 1381740423726556355*x^31 + 628063828966616525*x^29 + 245895771032252950*x^27 + 82199443459353129*x^25 + 23220181768178850*x^23 + 5475028879169850*x^21 + 1061639948869500*x^19 + 166245816169125*x^17 + 20554028180910*x^15 + 1949569068650*x^13 + 136627481900*x^11 + 6721387750*x^9 + 215659500*x^7 + 4025644*x^5 + 35720*x^3 + 95*x
96:
x^96 + 96*x^94 + 4464*x^92 + 133952*x^90 + 2915640*x^88 + 49059648*x^86 + 664122272*x^84 + 7432199808*x^82 + 70120271484*x^80 + 565977159360*x^78 + 3952626533856*x^76 + 24096225928320*x^74 + 129134734588080*x^72 + 611804229114240*x^70 + 2574054378590400*x^68 + 9652174278895360*x^66 + 32349865356610230*x^64 + 97121859358043520*x^62 + 261620279267892160*x^60 + 633038816547052800*x^58 + 1376859425989839840*x^56 + 2692525099713464576*x^54 + 4733419432051557504*x^52 + 7476490967123722752*x^50 + 10600348882785370800*x^48 + 13472894129331716352*x^46 + 15323566399844314752*x^44 + 15562097492488160768*x^42 + 14074544055706204224*x^40 + 11300200065312238080*x^38 + 8024283480721720576*x^36 + 5017666097126237184*x^34 + 2748936211413885801*x^32 + 1311661722810281248*x^30 + 541340464347696720*x^28 + 191687770982135232*x^26 + 57683819971475880*x^24 + 14586105645558720*x^22 + 3057523052744160*x^20 + 522653513289600*x^18 + 71398203154740*x^16 + 7598921843520*x^14 + 609790024480*x^12 + 35319123840*x^10 + 1389301200*x^8 + 33900160*x^6 + 442176*x^4 + 2304*x^2 + 2
97:
x^97 + 97*x^95 + 4559*x^93 + 138322*x^91 + 3045315*x^89 + 51849798*x^87 + 710513166*x^85 + 8052482548*x^83 + 76973590199*x^81 + 629783919810*x^79 + 4460607487206*x^77 + 27593525385972*x^75 + 150141241070730*x^73 + 722657841637140*x^71 + 3090885949170900*x^69 + 11790599084154360*x^67 + 40229960763989645*x^65 + 123056350572203620*x^63 + 338015545242635260*x^61 + 834775233190313800*x^59 + 1854935615569645340*x^57 + 3709871231139290680*x^55 + 6677768216050723224*x^53 + 10813119390972851472*x^51 + 15738273086176239300*x^49 + 20564676832603619352*x^47 + 24084957373829374344*x^45 + 25231860105916487408*x^43 + 23586304012052368664*x^41 + 19615384675337669680*x^39 + 14462686611368371376*x^37 + 9415590697137180192*x^35 + 5386804293073675206*x^33 + 2693402146536837603*x^31 + 1169404293314313245*x^29 + 437583542014388182*x^27 + 139883263430829009*x^25 + 37806287413737570*x^23 + 8532551931914010*x^21 + 1584293462159100*x^19 + 237644019323865*x^17 + 28152950024430*x^15 + 2559359093130*x^13 + 171946605740*x^11 + 8110688950*x^9 + 249559660*x^7 + 4467820*x^5 + 38024*x^3 + 97*x
98:
x^98 + 98*x^96 + 4655*x^94 + 142786*x^92 + 3179267*x^90 + 54765438*x^88 + 759572814*x^86 + 8716604820*x^84 + 84405790007*x^82 + 699904191294*x^80 + 5026584646566*x^78 + 31546151919828*x^76 + 174237466999050*x^74 + 851792576225220*x^72 + 3702690178285140*x^70 + 14364653462744760*x^68 + 49882135042885005*x^66 + 155406215928813850*x^64 + 435137404600678780*x^62 + 1096395512458205960*x^60 + 2487974432116698140*x^58 + 5086730657129130520*x^56 + 9370293315764187800*x^54 + 15546538823024408976*x^52 + 23214764053299962052*x^50 + 31165025715388990152*x^48 + 37557851503161090696*x^46 + 40555426505760802160*x^44 + 39148401504540529432*x^42 + 33689928731043873904*x^40 + 25762886676680609456*x^38 + 17439874177858900768*x^36 + 10404470390199912390*x^34 + 5442338357950723404*x^32 + 2481066016124594493*x^30 + 978924006362084902*x^28 + 331571034412964241*x^26 + 95490107385213450*x^24 + 23118657577472730*x^22 + 4641816514903260*x^20 + 760297532613465*x^18 + 99551153179170*x^16 + 10158280936650*x^14 + 781736630220*x^12 + 43429812790*x^10 + 1638860860*x^8 + 38367980*x^6 + 480200*x^4 + 2401*x^2 + 2
99:
x^99 + 99*x^97 + 4752*x^95 + 147345*x^93 + 3317589*x^91 + 57810753*x^89 + 811422612*x^87 + 9427117986*x^85 + 92458272555*x^83 + 776877781493*x^81 + 5656368566376*x^79 + 36006759407034*x^77 + 201830992385022*x^75 + 1001933817295950*x^73 + 4425348019922280*x^71 + 17455539411915660*x^69 + 61672734127039365*x^67 + 195636176692803495*x^65 + 558193755172882400*x^63 + 1434411057700841220*x^61 + 3322749665307011940*x^59 + 6941666272698775860*x^57 + 13080164546903478480*x^55 + 22224307039075132200*x^53 + 34027883444272813524*x^51 + 46903298801565229452*x^49 + 58122528335764710048*x^47 + 64640383879590176504*x^45 + 64380261610457016840*x^43 + 57276232743096242568*x^41 + 45378271352018279136*x^39 + 31902560789227272144*x^37 + 19820061087337092582*x^35 + 10829142651024398610*x^33 + 5174468162661432096*x^31 + 2148328299676398147*x^29 + 769154576427352423*x^27 + 235373370816042459*x^25 + 60924944991210300*x^23 + 13174368446817270*x^21 + 2344590994772565*x^19 + 337195172503035*x^17 + 38311230961080*x^15 + 3341095723350*x^13 + 215376418530*x^11 + 9749549810*x^9 + 287927640*x^7 + 4948020*x^5 + 40425*x^3 + 99*x
100:
x^100 + 100*x^98 + 4850*x^96 + 152000*x^94 + 3460375*x^92 + 60990020*x^90 + 866188050*x^88 + 10186690800*x^86 + 101174877375*x^84 + 861283571500*x^82 + 6356272757670*x^80 + 41033344053600*x^78 + 233377144304850*x^76 + 1176171284295000*x^74 + 5277140596147500*x^72 + 21158229590200800*x^70 + 76037387589784125*x^68 + 245518311735688500*x^66 + 713599971101696250*x^64 + 1869548462301520000*x^62 + 4419145177765217900*x^60 + 9429640704815474000*x^58 + 18166895204032609000*x^56 + 31594600354839320000*x^54 + 49574422267297222500*x^52 + 70118062854865191504*x^50 + 89287554051153700200*x^48 + 102198235382751267200*x^46 + 104935688116217819000*x^44 + 96424634247636772000*x^42 + 79068200083062153040*x^40 + 57665447465907881600*x^38 + 37259935265195993350*x^36 + 21233613041224311000*x^34 + 10616806520612155500*x^32 + 4629394315800992640*x^30 + 1748078582789437325*x^28 + 566944405229006700*x^26 + 156415052376423750*x^24 + 36293026024290000*x^22 + 6986407509675825*x^20 + 1097492705116500*x^18 + 137862384140250*x^16 + 13499376660000*x^14 + 997113048750*x^12 + 53179362600*x^10 + 1926788500*x^8 + 43316000*x^6 + 520625*x^4 + 2500*x^2 + 2
Definition
The Lucas polynomials $L_n$ are defined by $L_0 = 2$, $L_1 = x$ and $L_n(x) = x L_{n-1}(x) + L_{n-2}(x)$ for $n \geq 2$.
Parameters
$n$
—   integer ($n \geq 0$)
Formulas
(1)
$L_n(x) = F_{n-1}(x) + F_{n+1}(x)$ for $n \geq 1$, where $F_n$ are the Fibonacci polynomials.
(2)
$\sum_{n=0}^\infty L_n(x) t^n = \frac{2 - xt}{1 - xt - t^2}$.
(3)
$L_n(x) = \alpha^n + \beta^n$, where $\alpha, \beta = \frac{x \pm \sqrt{x^2+4}}{2}$ are the roots of $z^2 - xz - 1$.
(4)
$L_{2n}(x) = L_n(x)^2 - 2(-1)^n$.
(5)
$L_n(x)^2 - (x^2+4)F_n(x)^2 = 4(-1)^n$.
(6)
$L_n(x) = 2i^{n} T_{n}(-ix/2)$, where $T_n$ are the $T_n$.
Comments
(7)
At $x = 1$ the values are the Lucas numbers, and the ratio $L_{n+1}(1)/L_n(1)$ tends to the golden ratio.
(8)
Some authors index these from $L_1$, and elsewhere they are called the Fibonacci polynomials of the second kind.
(9)
They obey the same recurrence as the Fibonacci polynomials and differ only in the two starting values, which is why each table says which it holds.
(10)
For integers $P, Q$ the Lucas sequences [5] are the two solutions of $X_n = P X_{n-1} - Q X_{n-2}$ picked out by their starting values: $U_0 = 0$, $U_1 = 1$ and $V_0 = 2$, $V_1 = P$. The polynomials in this pair are the case $P = x$, $Q = -1$: $F_n(x) = U_n(x, -1)$ and $L_n(x) = V_n(x, -1)$.
(11)
The coefficients are the rows of [4].
(12)
[1] is a book-length treatment of both families and of the numbers they specialise to.
(13)
These are not orthogonal polynomials in the usual sense. A three-term recurrence $p_n = (x - c_n)p_{n-1} - k_n p_{n-2}$ gives a family orthogonal for a positive measure on $\mathbb{R}$ exactly when $k_n > 0$ (Favard), and here $k_n = -1$. What holds instead comes from $T_n$ through $L_n(x) = 2i^{n}T_{n}(-ix/2)$: along the segment $x = 2it$, $t \in [-1,1]$, with weight $1/\sqrt{1-t^2}$, distinct $L_m, L_n$ pair to zero, while $\langle L_n, L_n\rangle$ alternates between $\pm2\pi$ — an indefinite form rather than an inner product.
Programs
(P1)
Sage
R. = ZZ[]
def lucas_polynomial(n):
    a, b = R(2), x
    for _ in range(n):
        a, b = b, x*b + a
    return a

lucas_polynomial(101)            # the next one after this table
References
[1]
Koshy, T., "Fibonacci and Lucas Numbers with Applications", Wiley-Interscience, 2001.
Links
Data properties
Entries are of type: integral polynomial
Table is complete: no