Convolution Identities for the Pairs of k-Bonacci numbers up to , 10 By Shalosh B. Ekhad Theorem number, 1, Let a(n) be the, 2, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = ----------- / 2 ----- -x - x + 1 n = 0 and let b(n) be the, 2, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = ----------- / 2 ----- -x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- 2 \ n 2 x ) c(n) x = ------------------------ / 2 ----- (x - 1) (4 x + 2 x - 1) n = 0 and in Maple notation 2*x^2/(x-1)/(4*x^2+2*x-1) Theorem number, 2, Let a(n) be the, 2, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = ----------- / 2 ----- -x - x + 1 n = 0 and let b(n) be the, 3, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = ---------------- / 3 2 ----- -x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- 2 3 \ n x (3 x - 4 x + 2) ) c(n) x = -------------------------------------------- / 6 5 4 3 2 ----- 11 x - 11 x - 6 x + 7 x + 4 x - 5 x + 1 n = 0 and in Maple notation x^2*(3*x^3-4*x+2)/(11*x^6-11*x^5-6*x^4+7*x^3+4*x^2-5*x+1) Theorem number, 3, Let a(n) be the, 2, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = ----------- / 2 ----- -x - x + 1 n = 0 and let b(n) be the, 4, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = --------------------- / 4 3 2 ----- -x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- 2 4 3 2 \ n x (x - 1) (4 x + 5 x - 2 x - 4 x + 2) ) c(n) x = ----------------------------------------------------------- / 8 7 6 5 4 3 2 ----- 19 x - 25 x - 9 x + 6 x + 19 x - 8 x - 8 x + 6 x - 1 n = 0 and in Maple notation x^2*(x-1)*(4*x^4+5*x^3-2*x^2-4*x+2)/(19*x^8-25*x^7-9*x^6+6*x^5+19*x^4-8*x^3-8*x ^2+6*x-1) Theorem number, 4, Let a(n) be the, 2, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = ----------- / 2 ----- -x - x + 1 n = 0 and let b(n) be the, 5, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = -------------------------- / 5 4 3 2 ----- -x - x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n 2 7 6 5 4 3 2 ) c(n) x = x (6 x + 2 x - 7 x - 10 x + 11 x + 6 x - 8 x + 2) / ----- n = 0 / 10 9 8 7 6 5 4 3 2 / (36 x - 48 x + 4 x - 10 x + 41 x + 3 x - 35 x + 6 x + 13 x / - 7 x + 1) and in Maple notation x^2*(6*x^7+2*x^6-7*x^5-10*x^4+11*x^3+6*x^2-8*x+2)/(36*x^10-48*x^9+4*x^8-10*x^7+ 41*x^6+3*x^5-35*x^4+6*x^3+13*x^2-7*x+1) Theorem number, 5, Let a(n) be the, 2, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = ----------- / 2 ----- -x - x + 1 n = 0 and let b(n) be the, 6, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = ------------------------------- / 6 5 4 3 2 ----- -x - x - x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n 2 ) c(n) x = x / ----- n = 0 9 7 6 5 4 3 2 / 12 (9 x - 5 x - 23 x + 8 x + 27 x - 13 x - 12 x + 10 x - 2) / (59 x / 11 10 9 8 7 6 5 4 2 - 106 x + 48 x - 66 x + 75 x + 46 x - 71 x - 32 x + 54 x - 19 x + 8 x - 1) and in Maple notation x^2*(9*x^9-5*x^7-23*x^6+8*x^5+27*x^4-13*x^3-12*x^2+10*x-2)/(59*x^12-106*x^11+48 *x^10-66*x^9+75*x^8+46*x^7-71*x^6-32*x^5+54*x^4-19*x^2+8*x-1) Theorem number, 6, Let a(n) be the, 2, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = ----------- / 2 ----- -x - x + 1 n = 0 and let b(n) be the, 7, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = ------------------------------------ / 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n 2 11 10 9 8 7 6 5 ) c(n) x = x (14 x - 6 x + 6 x - 38 x - 5 x + 58 x + 6 x / ----- n = 0 4 3 2 / 14 13 12 - 52 x + 11 x + 20 x - 12 x + 2) / (101 x - 211 x + 196 x / 11 10 9 8 7 6 5 4 - 209 x + 143 x + 120 x - 91 x - 151 x + 93 x + 86 x - 73 x 3 2 - 11 x + 26 x - 9 x + 1) and in Maple notation x^2*(14*x^11-6*x^10+6*x^9-38*x^8-5*x^7+58*x^6+6*x^5-52*x^4+11*x^3+20*x^2-12*x+2 )/(101*x^14-211*x^13+196*x^12-209*x^11+143*x^10+120*x^9-91*x^8-151*x^7+93*x^6+ 86*x^5-73*x^4-11*x^3+26*x^2-9*x+1) Theorem number, 7, Let a(n) be the, 2, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = ----------- / 2 ----- -x - x + 1 n = 0 and let b(n) be the, 8, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = ----------------------------------------- / 8 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n 2 13 12 11 10 9 8 ) c(n) x = x (22 x - 22 x + 31 x - 66 x - 34 x + 85 x / ----- n = 0 7 6 5 4 3 2 / 16 + 69 x - 104 x - 47 x + 83 x - 3 x - 30 x + 14 x - 2) / (164 x / 15 14 13 12 11 10 9 - 430 x + 537 x - 611 x + 326 x + 189 x - 25 x - 372 x 8 7 6 5 4 3 2 + 35 x + 330 x - 80 x - 170 x + 88 x + 28 x - 34 x + 10 x - 1) and in Maple notation x^2*(22*x^13-22*x^12+31*x^11-66*x^10-34*x^9+85*x^8+69*x^7-104*x^6-47*x^5+83*x^4 -3*x^3-30*x^2+14*x-2)/(164*x^16-430*x^15+537*x^14-611*x^13+326*x^12+189*x^11-25 *x^10-372*x^9+35*x^8+330*x^7-80*x^6-170*x^5+88*x^4+28*x^3-34*x^2+10*x-1) Theorem number, 8, Let a(n) be the, 2, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = ----------- / 2 ----- -x - x + 1 n = 0 and let b(n) be the, 9, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = ---------------------------------------------- / 9 8 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n 2 15 14 13 12 11 10 ) c(n) x = x (35 x - 56 x + 92 x - 118 x - 61 x + 96 x / ----- n = 0 9 8 7 6 5 4 3 2 + 195 x - 120 x - 220 x + 140 x + 127 x - 116 x - 13 x + 42 x / 18 17 16 15 14 - 16 x + 2) / (271 x - 833 x + 1342 x - 1579 x + 956 x / 13 12 11 10 9 8 7 6 + 79 x + 253 x - 700 x - 301 x + 735 x + 215 x - 580 x - 2 x 5 4 3 2 + 286 x - 94 x - 52 x + 43 x - 11 x + 1) and in Maple notation x^2*(35*x^15-56*x^14+92*x^13-118*x^12-61*x^11+96*x^10+195*x^9-120*x^8-220*x^7+ 140*x^6+127*x^5-116*x^4-13*x^3+42*x^2-16*x+2)/(271*x^18-833*x^17+1342*x^16-1579 *x^15+956*x^14+79*x^13+253*x^12-700*x^11-301*x^10+735*x^9+215*x^8-580*x^7-2*x^6 +286*x^5-94*x^4-52*x^3+43*x^2-11*x+1) Theorem number, 9, Let a(n) be the, 2, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = ----------- / 2 ----- -x - x + 1 n = 0 and let b(n) be the, 10, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = ---------------------------------------------------- / 10 9 8 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n 2 17 16 15 14 13 12 ) c(n) x = x (56 x - 125 x + 225 x - 250 x - 61 x + 41 x / ----- n = 0 11 10 9 8 7 6 5 4 + 380 x - 29 x - 535 x + 40 x + 487 x - 129 x - 256 x + 145 x 3 2 / 20 19 18 17 + 39 x - 56 x + 18 x - 2) / (439 x - 1601 x + 3042 x - 3973 x / 16 15 14 13 12 11 + 2809 x - 875 x + 1103 x - 1108 x - 1112 x + 1122 x 10 9 8 7 6 5 4 3 + 1253 x - 1100 x - 797 x + 864 x + 194 x - 432 x + 85 x + 84 x 2 - 53 x + 12 x - 1) and in Maple notation x^2*(56*x^17-125*x^16+225*x^15-250*x^14-61*x^13+41*x^12+380*x^11-29*x^10-535*x^ 9+40*x^8+487*x^7-129*x^6-256*x^5+145*x^4+39*x^3-56*x^2+18*x-2)/(439*x^20-1601*x ^19+3042*x^18-3973*x^17+2809*x^16-875*x^15+1103*x^14-1108*x^13-1112*x^12+1122*x ^11+1253*x^10-1100*x^9-797*x^8+864*x^7+194*x^6-432*x^5+85*x^4+84*x^3-53*x^2+12* x-1) Theorem number, 10, Let a(n) be the, 3, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = ---------------- / 3 2 ----- -x - x - x + 1 n = 0 and let b(n) be the, 3, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = ---------------- / 3 2 ----- -x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- 2 2 \ n 2 x (2 x - 1) (x + x + 1) ) c(n) x = ---------------------------------------- / 3 3 2 ----- (2 x - 2 x + 1) (8 x + 4 x + 2 x - 1) n = 0 and in Maple notation 2*x^2*(2*x-1)*(x^2+x+1)/(2*x^3-2*x+1)/(8*x^3+4*x^2+2*x-1) Theorem number, 11, Let a(n) be the, 3, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = ---------------- / 3 2 ----- -x - x - x + 1 n = 0 and let b(n) be the, 4, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = --------------------- / 4 3 2 ----- -x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n 2 ) c(n) x = x / ----- n = 0 9 8 7 6 5 4 3 2 / (11 x + 23 x - 9 x - 15 x - 11 x + 4 x + 6 x + 6 x - 8 x + 2) / ( / 12 11 10 9 8 7 6 5 4 3 47 x + 90 x - 70 x - 193 x + 95 x + 68 x + x - 30 x - 9 x + x 2 + 13 x - 7 x + 1) and in Maple notation x^2*(11*x^9+23*x^8-9*x^7-15*x^6-11*x^5+4*x^4+6*x^3+6*x^2-8*x+2)/(47*x^12+90*x^ 11-70*x^10-193*x^9+95*x^8+68*x^7+x^6-30*x^5-9*x^4+x^3+13*x^2-7*x+1) Theorem number, 12, Let a(n) be the, 3, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = ---------------- / 3 2 ----- -x - x - x + 1 n = 0 and let b(n) be the, 5, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = -------------------------- / 5 4 3 2 ----- -x - x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n 2 12 11 10 9 8 7 6 ) c(n) x = x (9 x + 57 x + 39 x - 59 x - 25 x - 9 x + 12 x / ----- n = 0 5 3 2 / 15 14 13 + 26 x - 6 x - 12 x + 10 x - 2) / (54 x + 180 x + 270 x / 12 11 10 9 8 7 6 5 - 551 x - 272 x + 477 x + 58 x - 44 x - 56 x - 55 x + 38 x 4 3 2 + 16 x + 6 x - 19 x + 8 x - 1) and in Maple notation x^2*(9*x^12+57*x^11+39*x^10-59*x^9-25*x^8-9*x^7+12*x^6+26*x^5-6*x^3-12*x^2+10*x -2)/(54*x^15+180*x^14+270*x^13-551*x^12-272*x^11+477*x^10+58*x^9-44*x^8-56*x^7-\ 55*x^6+38*x^5+16*x^4+6*x^3-19*x^2+8*x-1) Theorem number, 13, Let a(n) be the, 3, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = ---------------- / 3 2 ----- -x - x - x + 1 n = 0 and let b(n) be the, 6, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = ------------------------------- / 6 5 4 3 2 ----- -x - x - x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n 2 15 14 13 12 11 10 ) c(n) x = x (x + 59 x + 202 x - 60 x - 204 x - 9 x / ----- n = 0 9 8 7 6 5 4 3 2 - 18 x + 52 x + 47 x + 10 x - 44 x - 8 x + 2 x + 20 x - 12 x + 2) / 18 17 16 15 14 13 12 / (53 x + 65 x + 1048 x - 188 x - 2311 x + 565 x + 1316 x / 11 10 9 8 7 6 5 4 - 379 x + 16 x - 213 x - 102 x + 73 x + 112 x - 35 x - 21 x 3 2 - 18 x + 26 x - 9 x + 1) and in Maple notation x^2*(x^15+59*x^14+202*x^13-60*x^12-204*x^11-9*x^10-18*x^9+52*x^8+47*x^7+10*x^6-\ 44*x^5-8*x^4+2*x^3+20*x^2-12*x+2)/(53*x^18+65*x^17+1048*x^16-188*x^15-2311*x^14 +565*x^13+1316*x^12-379*x^11+16*x^10-213*x^9-102*x^8+73*x^7+112*x^6-35*x^5-21*x ^4-18*x^3+26*x^2-9*x+1) Theorem number, 14, Let a(n) be the, 3, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = ---------------- / 3 2 ----- -x - x - x + 1 n = 0 and let b(n) be the, 7, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = ------------------------------------ / 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n 2 18 17 16 15 14 ) c(n) x = - 2 x (2 x + 12 x - 202 x - 224 x + 316 x / ----- n = 0 13 12 11 10 9 8 7 6 + 155 x - 78 x + 10 x - 88 x - 61 x + 18 x + 46 x + 30 x 5 4 3 2 / 21 20 19 - 29 x - 9 x - 4 x + 15 x - 7 x + 1) / (88 x - 440 x + 1712 x / 18 17 16 15 14 13 + 3112 x - 4448 x - 4968 x + 5976 x + 1252 x - 1704 x 12 11 10 9 8 7 6 5 + 596 x - 784 x - 280 x + 396 x + 232 x - 14 x - 186 x + 22 x 4 3 2 + 20 x + 36 x - 34 x + 10 x - 1) and in Maple notation -2*x^2*(2*x^18+12*x^17-202*x^16-224*x^15+316*x^14+155*x^13-78*x^12+10*x^11-88*x ^10-61*x^9+18*x^8+46*x^7+30*x^6-29*x^5-9*x^4-4*x^3+15*x^2-7*x+1)/(88*x^21-440*x ^20+1712*x^19+3112*x^18-4448*x^17-4968*x^16+5976*x^15+1252*x^14-1704*x^13+596*x ^12-784*x^11-280*x^10+396*x^9+232*x^8-14*x^7-186*x^6+22*x^5+20*x^4+36*x^3-34*x^ 2+10*x-1) Theorem number, 15, Let a(n) be the, 3, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = ---------------- / 3 2 ----- -x - x - x + 1 n = 0 and let b(n) be the, 8, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = ----------------------------------------- / 8 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n 2 21 20 19 18 17 16 ) c(n) x = x (9 x - 184 x + 236 x + 1772 x - 81 x - 2493 x / ----- n = 0 15 14 13 12 11 10 9 + 351 x + 327 x - 157 x + 561 x + 151 x - 163 x - 317 x 8 7 6 5 4 3 2 / - 54 x + 108 x + 144 x - 62 x - 26 x - 26 x + 42 x - 16 x + 2) / / 24 23 22 21 20 19 18 (221 x - 1312 x + 700 x + 8734 x + 3068 x - 23572 x + 644 x 17 16 15 14 13 12 + 20653 x - 7989 x - 1920 x + 2764 x - 3554 x + 36 x 11 10 9 8 7 6 4 3 + 1188 x + 763 x - 342 x - 413 x - 130 x + 264 x - 8 x - 61 x 2 + 43 x - 11 x + 1) and in Maple notation x^2*(9*x^21-184*x^20+236*x^19+1772*x^18-81*x^17-2493*x^16+351*x^15+327*x^14-157 *x^13+561*x^12+151*x^11-163*x^10-317*x^9-54*x^8+108*x^7+144*x^6-62*x^5-26*x^4-\ 26*x^3+42*x^2-16*x+2)/(221*x^24-1312*x^23+700*x^22+8734*x^21+3068*x^20-23572*x^ 19+644*x^18+20653*x^17-7989*x^16-1920*x^15+2764*x^14-3554*x^13+36*x^12+1188*x^ 11+763*x^10-342*x^9-413*x^8-130*x^7+264*x^6-8*x^4-61*x^3+43*x^2-11*x+1) Theorem number, 16, Let a(n) be the, 3, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = ---------------- / 3 2 ----- -x - x - x + 1 n = 0 and let b(n) be the, 9, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = ---------------------------------------------- / 9 8 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n 2 24 23 22 21 20 ) c(n) x = x (49 x - 283 x - 688 x + 2600 x + 5057 x / ----- n = 0 19 18 17 16 15 14 13 - 5595 x - 4913 x + 4423 x - 745 x + 161 x + 1783 x - 5 x 12 11 10 9 8 7 6 5 - 693 x - 727 x - 88 x + 510 x + 244 x - 52 x - 258 x + 52 x 4 3 2 / 27 26 25 + 26 x + 54 x - 56 x + 18 x - 2) / (514 x - 2180 x - 2448 x / 24 23 22 21 20 19 + 7831 x + 35020 x - 30306 x - 66870 x + 61394 x + 32050 x 18 17 16 15 14 13 - 45001 x + 16972 x + 1981 x - 12550 x + 3352 x + 3684 x 12 11 10 9 8 7 6 + 1469 x - 1014 x - 1672 x + 66 x + 547 x + 386 x - 333 x 5 4 3 2 - 26 x - 21 x + 94 x - 53 x + 12 x - 1) and in Maple notation x^2*(49*x^24-283*x^23-688*x^22+2600*x^21+5057*x^20-5595*x^19-4913*x^18+4423*x^ 17-745*x^16+161*x^15+1783*x^14-5*x^13-693*x^12-727*x^11-88*x^10+510*x^9+244*x^8 -52*x^7-258*x^6+52*x^5+26*x^4+54*x^3-56*x^2+18*x-2)/(514*x^27-2180*x^26-2448*x^ 25+7831*x^24+35020*x^23-30306*x^22-66870*x^21+61394*x^20+32050*x^19-45001*x^18+ 16972*x^17+1981*x^16-12550*x^15+3352*x^14+3684*x^13+1469*x^12-1014*x^11-1672*x^ 10+66*x^9+547*x^8+386*x^7-333*x^6-26*x^5-21*x^4+94*x^3-53*x^2+12*x-1) Theorem number, 17, Let a(n) be the, 3, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = ---------------- / 3 2 ----- -x - x - x + 1 n = 0 and let b(n) be the, 10, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = ---------------------------------------------------- / 10 9 8 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n 2 27 26 25 24 23 ) c(n) x = x (107 x - 134 x - 1798 x - 858 x + 15012 x / ----- n = 0 22 21 20 19 18 17 + 4531 x - 27472 x + 2662 x + 13168 x - 9088 x + 4254 x 16 15 14 13 12 11 10 + 3866 x - 1832 x - 2119 x - 1796 x + 464 x + 1375 x + 796 x 9 8 7 6 5 4 3 2 - 576 x - 502 x - 128 x + 390 x - 28 x - 10 x - 94 x + 72 x - 20 x / 30 29 28 27 26 + 2) / (991 x - 2831 x - 3430 x - 15663 x + 74916 x / 25 24 23 22 21 20 + 67713 x - 218155 x - 58865 x + 278803 x - 86224 x - 102342 x 19 18 17 16 15 14 + 108360 x - 42106 x - 27667 x + 20588 x + 7113 x + 3336 x 13 12 11 10 9 8 7 - 4733 x - 4294 x - 19 x + 2668 x + 534 x - 520 x - 766 x 6 5 4 3 2 + 380 x + 46 x + 74 x - 136 x + 64 x - 13 x + 1) and in Maple notation x^2*(107*x^27-134*x^26-1798*x^25-858*x^24+15012*x^23+4531*x^22-27472*x^21+2662* x^20+13168*x^19-9088*x^18+4254*x^17+3866*x^16-1832*x^15-2119*x^14-1796*x^13+464 *x^12+1375*x^11+796*x^10-576*x^9-502*x^8-128*x^7+390*x^6-28*x^5-10*x^4-94*x^3+ 72*x^2-20*x+2)/(991*x^30-2831*x^29-3430*x^28-15663*x^27+74916*x^26+67713*x^25-\ 218155*x^24-58865*x^23+278803*x^22-86224*x^21-102342*x^20+108360*x^19-42106*x^ 18-27667*x^17+20588*x^16+7113*x^15+3336*x^14-4733*x^13-4294*x^12-19*x^11+2668*x ^10+534*x^9-520*x^8-766*x^7+380*x^6+46*x^5+74*x^4-136*x^3+64*x^2-13*x+1) Theorem number, 18, Let a(n) be the, 4, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = --------------------- / 4 3 2 ----- -x - x - x - x + 1 n = 0 and let b(n) be the, 4, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = --------------------- / 4 3 2 ----- -x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n ) c(n) x = / ----- n = 0 2 7 6 5 4 3 2 2 x (4 x + 6 x + 8 x - 2 x - x - x - 2 x + 1) ------------------------------------------------------------------------ 4 3 2 6 5 4 3 2 (16 x + 8 x + 4 x + 2 x - 1) (x + 6 x - 4 x - 3 x - x + 3 x - 1) and in Maple notation 2*x^2*(4*x^7+6*x^6+8*x^5-2*x^4-x^3-x^2-2*x+1)/(16*x^4+8*x^3+4*x^2+2*x-1)/(x^6+6 *x^5-4*x^4-3*x^3-x^2+3*x-1) Theorem number, 19, Let a(n) be the, 4, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = --------------------- / 4 3 2 ----- -x - x - x - x + 1 n = 0 and let b(n) be the, 5, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = -------------------------- / 5 4 3 2 ----- -x - x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n 2 17 16 15 14 13 12 ) c(n) x = - x (8 x - 38 x - 171 x - 360 x - 40 x + 306 x / ----- n = 0 11 10 9 8 7 6 5 4 3 + 442 x - 202 x - 120 x - 56 x - 3 x + 27 x + 13 x + 14 x - 2 x 2 / 20 19 18 17 16 - 20 x + 12 x - 2) / (29 x - 424 x + 1453 x + 2133 x - 1096 x / 15 14 13 12 11 10 9 - 3753 x - 319 x + 2137 x + 1872 x - 1452 x - 292 x - 103 x 8 7 6 5 4 3 2 + 99 x + 86 x + 10 x + 8 x - 27 x - 18 x + 26 x - 9 x + 1) and in Maple notation -x^2*(8*x^17-38*x^16-171*x^15-360*x^14-40*x^13+306*x^12+442*x^11-202*x^10-120*x ^9-56*x^8-3*x^7+27*x^6+13*x^5+14*x^4-2*x^3-20*x^2+12*x-2)/(29*x^20-424*x^19+ 1453*x^18+2133*x^17-1096*x^16-3753*x^15-319*x^14+2137*x^13+1872*x^12-1452*x^11-\ 292*x^10-103*x^9+99*x^8+86*x^7+10*x^6+8*x^5-27*x^4-18*x^3+26*x^2-9*x+1) Theorem number, 20, Let a(n) be the, 4, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = --------------------- / 4 3 2 ----- -x - x - x - x + 1 n = 0 and let b(n) be the, 6, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = ------------------------------- / 6 5 4 3 2 ----- -x - x - x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n 2 21 20 19 18 17 16 ) c(n) x = x (7 x - 114 x - 30 x + 631 x + 1114 x + 1103 x / ----- n = 0 15 14 13 12 11 10 9 - 1779 x - 1676 x - 214 x + 1384 x + 27 x + 4 x - 44 x 8 7 6 5 4 3 2 / - 83 x - 65 x + 22 x + 9 x + 26 x + 8 x - 30 x + 14 x - 2) / ( / 24 23 22 21 20 19 18 311 x - 1044 x - 2509 x + 6247 x + 9982 x - 3989 x - 12412 x 17 16 15 14 13 12 - 11812 x + 12960 x + 8281 x - 1332 x - 6136 x + 1386 x 11 10 9 8 7 6 5 4 + 26 x + 250 x + 230 x + 92 x - 143 x - 17 x - 36 x + 27 x 3 2 + 36 x - 34 x + 10 x - 1) and in Maple notation x^2*(7*x^21-114*x^20-30*x^19+631*x^18+1114*x^17+1103*x^16-1779*x^15-1676*x^14-\ 214*x^13+1384*x^12+27*x^11+4*x^10-44*x^9-83*x^8-65*x^7+22*x^6+9*x^5+26*x^4+8*x^ 3-30*x^2+14*x-2)/(311*x^24-1044*x^23-2509*x^22+6247*x^21+9982*x^20-3989*x^19-\ 12412*x^18-11812*x^17+12960*x^16+8281*x^15-1332*x^14-6136*x^13+1386*x^12+26*x^ 11+250*x^10+230*x^9+92*x^8-143*x^7-17*x^6-36*x^5+27*x^4+36*x^3-34*x^2+10*x-1) Theorem number, 21, Let a(n) be the, 4, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = --------------------- / 4 3 2 ----- -x - x - x - x + 1 n = 0 and let b(n) be the, 7, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = ------------------------------------ / 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n 2 25 24 23 22 21 20 ) c(n) x = x (84 x - 13 x - 873 x - 302 x + 708 x + 5622 x / ----- n = 0 19 18 17 16 15 14 13 + 4687 x - 2604 x - 11298 x - 6 x + 4618 x + 3332 x - 1801 x 12 11 10 9 8 7 6 5 + 107 x - 291 x - 257 x - 98 x + 83 x + 131 x - 9 x + 9 x 4 3 2 / 28 27 26 - 36 x - 26 x + 42 x - 16 x + 2) / (667 x + 1520 x - 8084 x / 25 24 23 22 21 20 - 13613 x + 27971 x + 30348 x + 15281 x - 56350 x - 54915 x 19 18 17 16 15 14 + 23767 x + 73351 x - 13525 x - 23598 x - 8102 x + 10240 x 13 12 11 10 9 8 7 - 2689 x + 1366 x + 536 x + 53 x - 485 x - 228 x + 149 x 6 5 4 3 2 + 10 x + 75 x - 16 x - 61 x + 43 x - 11 x + 1) and in Maple notation x^2*(84*x^25-13*x^24-873*x^23-302*x^22+708*x^21+5622*x^20+4687*x^19-2604*x^18-\ 11298*x^17-6*x^16+4618*x^15+3332*x^14-1801*x^13+107*x^12-291*x^11-257*x^10-98*x ^9+83*x^8+131*x^7-9*x^6+9*x^5-36*x^4-26*x^3+42*x^2-16*x+2)/(667*x^28+1520*x^27-\ 8084*x^26-13613*x^25+27971*x^24+30348*x^23+15281*x^22-56350*x^21-54915*x^20+ 23767*x^19+73351*x^18-13525*x^17-23598*x^16-8102*x^15+10240*x^14-2689*x^13+1366 *x^12+536*x^11+53*x^10-485*x^9-228*x^8+149*x^7+10*x^6+75*x^5-16*x^4-61*x^3+43*x ^2-11*x+1) Theorem number, 22, Let a(n) be the, 4, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = --------------------- / 4 3 2 ----- -x - x - x - x + 1 n = 0 and let b(n) be the, 8, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = ----------------------------------------- / 8 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n 2 29 28 27 26 25 ) c(n) x = x (111 x + 802 x - 438 x - 3081 x - 3850 x / ----- n = 0 24 23 22 21 20 19 - 416 x + 15707 x + 32627 x - 9441 x - 39821 x - 26116 x 18 17 16 15 14 13 + 32031 x + 12825 x - 2564 x - 7186 x + 2486 x - 1581 x 12 11 10 9 8 7 6 5 - 572 x + 271 x + 384 x + 307 x + 10 x - 193 x - 2 x - 45 x 4 3 2 / 32 31 30 + 38 x + 54 x - 56 x + 18 x - 2) / (523 x + 4669 x + 11482 x / 29 28 27 26 25 24 - 52599 x - 39433 x + 67775 x + 118479 x + 132450 x - 141235 x 23 22 21 20 19 - 385515 x + 79520 x + 287256 x + 103651 x - 208157 x 18 17 16 15 14 13 - 18142 x + 29335 x + 22206 x - 17745 x + 9947 x + 533 x 12 11 10 9 8 7 6 - 1566 x - 1276 x - 414 x + 455 x + 450 x - 141 x + 31 x 5 4 3 2 - 120 x - 12 x + 94 x - 53 x + 12 x - 1) and in Maple notation x^2*(111*x^29+802*x^28-438*x^27-3081*x^26-3850*x^25-416*x^24+15707*x^23+32627*x ^22-9441*x^21-39821*x^20-26116*x^19+32031*x^18+12825*x^17-2564*x^16-7186*x^15+ 2486*x^14-1581*x^13-572*x^12+271*x^11+384*x^10+307*x^9+10*x^8-193*x^7-2*x^6-45* x^5+38*x^4+54*x^3-56*x^2+18*x-2)/(523*x^32+4669*x^31+11482*x^30-52599*x^29-\ 39433*x^28+67775*x^27+118479*x^26+132450*x^25-141235*x^24-385515*x^23+79520*x^ 22+287256*x^21+103651*x^20-208157*x^19-18142*x^18+29335*x^17+22206*x^16-17745*x ^15+9947*x^14+533*x^13-1566*x^12-1276*x^11-414*x^10+455*x^9+450*x^8-141*x^7+31* x^6-120*x^5-12*x^4+94*x^3-53*x^2+12*x-1) Theorem number, 23, Let a(n) be the, 4, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = --------------------- / 4 3 2 ----- -x - x - x - x + 1 n = 0 and let b(n) be the, 9, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = ---------------------------------------------- / 9 8 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n 2 33 32 31 30 29 ) c(n) x = - x (16 x - 1016 x - 4864 x + 1504 x + 8210 x / ----- n = 0 28 27 26 25 24 23 + 15762 x + 35624 x - 58072 x - 128737 x - 52597 x + 203494 x 22 21 20 19 18 17 + 119039 x - 72112 x - 147879 x + 39688 x + 27222 x + 4352 x 16 15 14 13 12 11 10 - 9764 x + 8927 x - 733 x - 1862 x - 1366 x - 230 x + 261 x 9 8 7 6 5 4 3 2 + 534 x + 212 x - 238 x + 7 x - 99 x + 24 x + 94 x - 72 x + 20 x / 36 35 34 33 32 - 2) / (592 x - 3696 x + 47960 x + 40984 x - 222432 x / 31 30 29 28 27 - 103936 x + 42060 x + 288094 x + 1082089 x - 476722 x 26 25 24 23 22 - 1589504 x - 462617 x + 1666871 x + 586656 x - 629616 x 21 20 19 18 17 16 - 646542 x + 396796 x + 53856 x - 31758 x - 37854 x + 49451 x 15 14 13 12 11 10 - 12678 x - 6812 x - 3401 x + 1124 x + 1386 x + 1248 x 9 8 7 6 5 4 3 2 - 239 x - 692 x + 134 x - 122 x + 161 x + 64 x - 136 x + 64 x - 13 x + 1) and in Maple notation -x^2*(16*x^33-1016*x^32-4864*x^31+1504*x^30+8210*x^29+15762*x^28+35624*x^27-\ 58072*x^26-128737*x^25-52597*x^24+203494*x^23+119039*x^22-72112*x^21-147879*x^ 20+39688*x^19+27222*x^18+4352*x^17-9764*x^16+8927*x^15-733*x^14-1862*x^13-1366* x^12-230*x^11+261*x^10+534*x^9+212*x^8-238*x^7+7*x^6-99*x^5+24*x^4+94*x^3-72*x^ 2+20*x-2)/(592*x^36-3696*x^35+47960*x^34+40984*x^33-222432*x^32-103936*x^31+ 42060*x^30+288094*x^29+1082089*x^28-476722*x^27-1589504*x^26-462617*x^25+ 1666871*x^24+586656*x^23-629616*x^22-646542*x^21+396796*x^20+53856*x^19-31758*x ^18-37854*x^17+49451*x^16-12678*x^15-6812*x^14-3401*x^13+1124*x^12+1386*x^11+ 1248*x^10-239*x^9-692*x^8+134*x^7-122*x^6+161*x^5+64*x^4-136*x^3+64*x^2-13*x+1) Theorem number, 24, Let a(n) be the, 4, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = --------------------- / 4 3 2 ----- -x - x - x - x + 1 n = 0 and let b(n) be the, 10, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = ---------------------------------------------------- / 10 9 8 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n 2 37 36 35 34 33 ) c(n) x = x (4 x - 1554 x + 7536 x + 21735 x - 4874 x / ----- n = 0 32 31 30 29 28 27 + 18315 x - 91643 x - 204032 x + 53340 x + 650520 x + 343665 x 26 25 24 23 22 - 571425 x - 953778 x + 337489 x + 606300 x + 124022 x 21 20 19 18 17 16 - 382957 x + 40565 x + 32985 x - 758 x - 29852 x + 16710 x 15 14 13 12 11 10 9 + 5742 x + 2552 x - 798 x - 2159 x - 995 x - 258 x + 652 x 8 7 6 5 4 3 2 + 518 x - 264 x + 60 x - 165 x - 16 x + 148 x - 90 x + 22 x - 2) / 40 39 38 37 36 / (3043 x - 18672 x - 13995 x + 299501 x + 85397 x / 35 34 33 32 31 - 721925 x - 844 x - 1469905 x + 1092267 x + 5283960 x 30 29 28 27 26 + 318502 x - 7792891 x - 3453352 x + 5245859 x + 6238755 x 25 24 23 22 21 - 3603164 x - 3072092 x + 430743 x + 1905972 x - 769362 x 20 19 18 17 16 15 - 89270 x + 88297 x + 127177 x - 101394 x - 5721 x - 4164 x 14 13 12 11 10 9 8 + 7500 x + 6434 x + 871 x - 977 x - 2163 x - 280 x + 929 x 7 6 5 4 3 2 - 167 x + 275 x - 182 x - 148 x + 188 x - 76 x + 14 x - 1) and in Maple notation x^2*(4*x^37-1554*x^36+7536*x^35+21735*x^34-4874*x^33+18315*x^32-91643*x^31-\ 204032*x^30+53340*x^29+650520*x^28+343665*x^27-571425*x^26-953778*x^25+337489*x ^24+606300*x^23+124022*x^22-382957*x^21+40565*x^20+32985*x^19-758*x^18-29852*x^ 17+16710*x^16+5742*x^15+2552*x^14-798*x^13-2159*x^12-995*x^11-258*x^10+652*x^9+ 518*x^8-264*x^7+60*x^6-165*x^5-16*x^4+148*x^3-90*x^2+22*x-2)/(3043*x^40-18672*x ^39-13995*x^38+299501*x^37+85397*x^36-721925*x^35-844*x^34-1469905*x^33+1092267 *x^32+5283960*x^31+318502*x^30-7792891*x^29-3453352*x^28+5245859*x^27+6238755*x ^26-3603164*x^25-3072092*x^24+430743*x^23+1905972*x^22-769362*x^21-89270*x^20+ 88297*x^19+127177*x^18-101394*x^17-5721*x^16-4164*x^15+7500*x^14+6434*x^13+871* x^12-977*x^11-2163*x^10-280*x^9+929*x^8-167*x^7+275*x^6-182*x^5-148*x^4+188*x^3 -76*x^2+14*x-1) Theorem number, 25, Let a(n) be the, 5, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = -------------------------- / 5 4 3 2 ----- -x - x - x - x - x + 1 n = 0 and let b(n) be the, 5, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = -------------------------- / 5 4 3 2 ----- -x - x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n 2 10 9 8 6 4 3 ) c(n) x = - 2 x (52 x + 40 x + 19 x - 25 x - x - x + 3 x - 1) / ----- n = 0 / 5 4 3 2 / ((32 x + 16 x + 8 x + 4 x + 2 x - 1) / 10 9 8 7 6 4 3 2 (4 x - 4 x - 15 x - 12 x + 25 x - 2 x - 4 x - 3 x + 4 x - 1)) and in Maple notation -2*x^2*(52*x^10+40*x^9+19*x^8-25*x^6-x^4-x^3+3*x-1)/(32*x^5+16*x^4+8*x^3+4*x^2+ 2*x-1)/(4*x^10-4*x^9-15*x^8-12*x^7+25*x^6-2*x^4-4*x^3-3*x^2+4*x-1) Theorem number, 26, Let a(n) be the, 5, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = -------------------------- / 5 4 3 2 ----- -x - x - x - x - x + 1 n = 0 and let b(n) be the, 6, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = ------------------------------- / 6 5 4 3 2 ----- -x - x - x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n 2 27 26 25 24 23 ) c(n) x = x (50 x + 379 x + 225 x - 2056 x - 6820 x / ----- n = 0 22 21 20 19 18 17 + 3694 x + 13812 x + 11293 x - 4285 x - 16073 x - 5837 x 16 15 14 13 12 11 10 + 2148 x + 6758 x + 4421 x - 4367 x - 201 x - 153 x - 13 x 9 8 7 6 5 4 3 2 + 71 x + 58 x + 14 x + 44 x + 2 x - 36 x - 26 x + 42 x - 16 x + 2) / 30 29 28 27 26 25 / (733 x + 3859 x + 2975 x - 23513 x - 43881 x + 13627 x / 24 23 22 21 20 19 + 168905 x + 17932 x - 127279 x - 114747 x + 35175 x + 96467 x 18 17 16 15 14 13 + 15540 x - 18372 x - 30442 x - 6491 x + 18959 x - 3945 x 12 11 10 9 8 7 6 5 + 711 x - 124 x - 361 x - 97 x - 19 x - 71 x + 74 x + 68 x 4 3 2 - 16 x - 61 x + 43 x - 11 x + 1) and in Maple notation x^2*(50*x^27+379*x^26+225*x^25-2056*x^24-6820*x^23+3694*x^22+13812*x^21+11293*x ^20-4285*x^19-16073*x^18-5837*x^17+2148*x^16+6758*x^15+4421*x^14-4367*x^13-201* x^12-153*x^11-13*x^10+71*x^9+58*x^8+14*x^7+44*x^6+2*x^5-36*x^4-26*x^3+42*x^2-16 *x+2)/(733*x^30+3859*x^29+2975*x^28-23513*x^27-43881*x^26+13627*x^25+168905*x^ 24+17932*x^23-127279*x^22-114747*x^21+35175*x^20+96467*x^19+15540*x^18-18372*x^ 17-30442*x^16-6491*x^15+18959*x^14-3945*x^13+711*x^12-124*x^11-361*x^10-97*x^9-\ 19*x^8-71*x^7+74*x^6+68*x^5-16*x^4-61*x^3+43*x^2-11*x+1) Theorem number, 27, Let a(n) be the, 5, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = -------------------------- / 5 4 3 2 ----- -x - x - x - x - x + 1 n = 0 and let b(n) be the, 7, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = ------------------------------------ / 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n 2 32 31 30 29 28 ) c(n) x = x (25 x + 621 x + 1975 x + 9203 x - 1913 x / ----- n = 0 27 26 25 24 23 22 - 36083 x - 51702 x + 43862 x + 96438 x + 42496 x - 32346 x 21 20 19 18 17 16 - 73064 x - 58434 x + 45484 x + 35652 x + 13972 x - 10843 x 15 14 13 12 11 10 9 - 17565 x + 7089 x - 465 x + 70 x + 268 x + 243 x + 87 x 8 7 6 5 4 3 2 / - 83 x + 13 x - 78 x - 36 x + 38 x + 54 x - 56 x + 18 x - 2) / ( / 35 34 33 32 31 30 46 x + 2106 x + 25872 x + 42402 x + 44216 x - 373946 x 29 28 27 26 25 - 356148 x + 477981 x + 1064044 x - 223641 x - 878244 x 24 23 22 21 20 - 377784 x + 226544 x + 459543 x + 267288 x - 305779 x 19 18 17 16 15 14 - 115026 x - 17994 x + 67672 x + 41319 x - 38316 x + 9847 x 13 12 11 10 9 8 7 6 - 852 x - 1035 x - 480 x - 8 x + 356 x - 35 x + 184 x - 51 x 5 4 3 2 - 112 x - 12 x + 94 x - 53 x + 12 x - 1) and in Maple notation x^2*(25*x^32+621*x^31+1975*x^30+9203*x^29-1913*x^28-36083*x^27-51702*x^26+43862 *x^25+96438*x^24+42496*x^23-32346*x^22-73064*x^21-58434*x^20+45484*x^19+35652*x ^18+13972*x^17-10843*x^16-17565*x^15+7089*x^14-465*x^13+70*x^12+268*x^11+243*x^ 10+87*x^9-83*x^8+13*x^7-78*x^6-36*x^5+38*x^4+54*x^3-56*x^2+18*x-2)/(46*x^35+ 2106*x^34+25872*x^33+42402*x^32+44216*x^31-373946*x^30-356148*x^29+477981*x^28+ 1064044*x^27-223641*x^26-878244*x^25-377784*x^24+226544*x^23+459543*x^22+267288 *x^21-305779*x^20-115026*x^19-17994*x^18+67672*x^17+41319*x^16-38316*x^15+9847* x^14-852*x^13-1035*x^12-480*x^11-8*x^10+356*x^9-35*x^8+184*x^7-51*x^6-112*x^5-\ 12*x^4+94*x^3-53*x^2+12*x-1) Theorem number, 28, Let a(n) be the, 5, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = -------------------------- / 5 4 3 2 ----- -x - x - x - x - x + 1 n = 0 and let b(n) be the, 8, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = ----------------------------------------- / 8 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n 2 37 36 35 34 33 ) c(n) x = - x (78 x - 484 x + 389 x - 4611 x - 42362 x / ----- n = 0 32 31 30 29 28 - 125418 x + 84212 x + 384175 x + 233473 x - 388815 x 27 26 25 24 23 - 478427 x - 283391 x + 273999 x + 432518 x + 263847 x 22 21 20 19 18 17 - 153644 x - 365882 x + 26486 x + 73892 x + 66120 x + 4783 x 16 15 14 13 12 11 - 40241 x + 13529 x - 1986 x - 1130 x - 532 x + 140 x 10 9 8 7 6 5 4 3 2 + 423 x + 212 x - 74 x + 91 x - 96 x - 88 x + 24 x + 94 x - 72 x / 40 39 38 37 36 + 20 x - 2) / (772 x - 6332 x + 9760 x - 21494 x + 286627 x / 35 34 33 32 31 + 969378 x - 258169 x - 4345110 x - 1040728 x + 5485464 x 30 29 28 27 26 + 4759245 x - 2880877 x - 3897554 x - 2667689 x + 1859275 x 25 24 23 22 21 + 2511006 x + 1289187 x - 1319387 x - 1552247 x + 567933 x 20 19 18 17 16 15 + 275159 x + 166460 x - 85562 x - 114639 x + 81141 x - 22042 x 14 13 12 11 10 9 8 - 2128 x - 908 x + 1048 x + 1410 x + 72 x - 330 x + 133 x 7 6 5 4 3 2 - 318 x - 20 x + 152 x + 64 x - 136 x + 64 x - 13 x + 1) and in Maple notation -x^2*(78*x^37-484*x^36+389*x^35-4611*x^34-42362*x^33-125418*x^32+84212*x^31+ 384175*x^30+233473*x^29-388815*x^28-478427*x^27-283391*x^26+273999*x^25+432518* x^24+263847*x^23-153644*x^22-365882*x^21+26486*x^20+73892*x^19+66120*x^18+4783* x^17-40241*x^16+13529*x^15-1986*x^14-1130*x^13-532*x^12+140*x^11+423*x^10+212*x ^9-74*x^8+91*x^7-96*x^6-88*x^5+24*x^4+94*x^3-72*x^2+20*x-2)/(772*x^40-6332*x^39 +9760*x^38-21494*x^37+286627*x^36+969378*x^35-258169*x^34-4345110*x^33-1040728* x^32+5485464*x^31+4759245*x^30-2880877*x^29-3897554*x^28-2667689*x^27+1859275*x ^26+2511006*x^25+1289187*x^24-1319387*x^23-1552247*x^22+567933*x^21+275159*x^20 +166460*x^19-85562*x^18-114639*x^17+81141*x^16-22042*x^15-2128*x^14-908*x^13+ 1048*x^12+1410*x^11+72*x^10-330*x^9+133*x^8-318*x^7-20*x^6+152*x^5+64*x^4-136*x ^3+64*x^2-13*x+1) Theorem number, 29, Let a(n) be the, 5, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = -------------------------- / 5 4 3 2 ----- -x - x - x - x - x + 1 n = 0 and let b(n) be the, 9, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = ---------------------------------------------- / 9 8 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n 2 42 41 40 39 38 ) c(n) x = - x (63 x + 203 x - 3955 x - 259 x + 111304 x / ----- n = 0 37 36 35 34 33 - 54312 x - 758588 x - 1092214 x + 1429001 x + 2675979 x 32 31 30 29 28 + 725801 x - 2484809 x - 2378950 x - 1582684 x + 1219923 x 27 26 25 24 23 + 3363373 x + 926697 x - 1184321 x - 1745396 x - 459272 x 22 21 20 19 18 + 1124791 x + 145441 x - 35433 x - 137299 x - 53540 x 17 16 15 14 13 12 + 81550 x - 30933 x - 705 x + 1101 x + 1395 x + 1243 x 11 10 9 8 7 6 5 4 + 333 x - 360 x - 354 x + 99 x - 229 x + 74 x + 152 x + 16 x 3 2 / 45 44 43 - 148 x + 90 x - 22 x + 2) / (2242 x - 563 x + 8830 x / 42 41 40 39 38 + 60619 x - 791896 x - 1386532 x + 7852392 x + 9079786 x 37 36 35 34 33 - 13206364 x - 32335616 x + 9811544 x + 35428768 x + 16800448 x 32 31 30 29 28 - 18663198 x - 17478938 x - 16541137 x + 8396636 x + 21889896 x 27 26 25 24 23 + 2864726 x - 7955847 x - 7829364 x + 721874 x + 5210378 x 22 21 20 19 18 - 812513 x - 320036 x - 452560 x + 64668 x + 275992 x 17 16 15 14 13 12 - 179938 x + 27959 x + 3776 x + 4184 x + 3510 x - 765 x 11 10 9 8 7 6 5 4 - 1534 x - 511 x + 346 x - 391 x + 442 x + 151 x - 172 x - 148 x 3 2 + 188 x - 76 x + 14 x - 1) and in Maple notation -x^2*(63*x^42+203*x^41-3955*x^40-259*x^39+111304*x^38-54312*x^37-758588*x^36-\ 1092214*x^35+1429001*x^34+2675979*x^33+725801*x^32-2484809*x^31-2378950*x^30-\ 1582684*x^29+1219923*x^28+3363373*x^27+926697*x^26-1184321*x^25-1745396*x^24-\ 459272*x^23+1124791*x^22+145441*x^21-35433*x^20-137299*x^19-53540*x^18+81550*x^ 17-30933*x^16-705*x^15+1101*x^14+1395*x^13+1243*x^12+333*x^11-360*x^10-354*x^9+ 99*x^8-229*x^7+74*x^6+152*x^5+16*x^4-148*x^3+90*x^2-22*x+2)/(2242*x^45-563*x^44 +8830*x^43+60619*x^42-791896*x^41-1386532*x^40+7852392*x^39+9079786*x^38-\ 13206364*x^37-32335616*x^36+9811544*x^35+35428768*x^34+16800448*x^33-18663198*x ^32-17478938*x^31-16541137*x^30+8396636*x^29+21889896*x^28+2864726*x^27-7955847 *x^26-7829364*x^25+721874*x^24+5210378*x^23-812513*x^22-320036*x^21-452560*x^20 +64668*x^19+275992*x^18-179938*x^17+27959*x^16+3776*x^15+4184*x^14+3510*x^13-\ 765*x^12-1534*x^11-511*x^10+346*x^9-391*x^8+442*x^7+151*x^6-172*x^5-148*x^4+188 *x^3-76*x^2+14*x-1) Theorem number, 30, Let a(n) be the, 5, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = -------------------------- / 5 4 3 2 ----- -x - x - x - x - x + 1 n = 0 and let b(n) be the, 10, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = ---------------------------------------------------- / 10 9 8 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n 2 47 46 45 44 43 ) c(n) x = x (114 x - 763 x - 5023 x + 84859 x + 66841 x / ----- n = 0 42 41 40 39 38 - 595029 x - 2105870 x + 2016757 x + 9074885 x + 5304408 x 37 36 35 34 33 - 14727079 x - 13796941 x - 1234687 x + 10773972 x + 16057923 x 32 31 30 29 28 + 9730718 x - 9007651 x - 18699187 x - 8051935 x + 11807162 x 27 26 25 24 23 + 8669042 x + 1149621 x - 4933647 x - 3717931 x + 2405957 x 22 21 20 19 18 + 402798 x + 142140 x - 202219 x - 149799 x + 183652 x 17 16 15 14 13 12 - 48869 x - 7405 x - 4570 x - 1013 x + 303 x + 1680 x 11 10 9 8 7 6 5 4 + 1117 x - 100 x - 466 x + 234 x - 413 x - 14 x + 214 x + 94 x 3 2 / 50 49 48 - 218 x + 110 x - 24 x + 2) / (1233 x + 11859 x - 14912 x / 47 46 45 44 43 + 145029 x + 2809451 x - 6164844 x - 20390779 x + 8591698 x 42 41 40 39 + 98726123 x + 31023599 x - 165573172 x - 164596640 x 38 37 36 35 + 150221775 x + 155903448 x + 57663700 x - 55834862 x 34 33 32 31 - 127637780 x - 100327856 x + 63965456 x + 126067526 x 30 29 28 27 26 + 29898405 x - 79341664 x - 31765036 x + 6260490 x + 25727289 x 25 24 23 22 21 + 7049175 x - 13563881 x + 1454223 x + 82573 x + 957977 x 20 19 18 17 16 15 + 51565 x - 692831 x + 325486 x - 22888 x + 11530 x + 204 x 14 13 12 11 10 9 8 - 3849 x - 5437 x - 1289 x + 1631 x + 882 x - 464 x + 852 x 7 6 5 4 3 2 - 499 x - 345 x + 149 x + 273 x - 251 x + 89 x - 15 x + 1) and in Maple notation x^2*(114*x^47-763*x^46-5023*x^45+84859*x^44+66841*x^43-595029*x^42-2105870*x^41 +2016757*x^40+9074885*x^39+5304408*x^38-14727079*x^37-13796941*x^36-1234687*x^ 35+10773972*x^34+16057923*x^33+9730718*x^32-9007651*x^31-18699187*x^30-8051935* x^29+11807162*x^28+8669042*x^27+1149621*x^26-4933647*x^25-3717931*x^24+2405957* x^23+402798*x^22+142140*x^21-202219*x^20-149799*x^19+183652*x^18-48869*x^17-\ 7405*x^16-4570*x^15-1013*x^14+303*x^13+1680*x^12+1117*x^11-100*x^10-466*x^9+234 *x^8-413*x^7-14*x^6+214*x^5+94*x^4-218*x^3+110*x^2-24*x+2)/(1233*x^50+11859*x^ 49-14912*x^48+145029*x^47+2809451*x^46-6164844*x^45-20390779*x^44+8591698*x^43+ 98726123*x^42+31023599*x^41-165573172*x^40-164596640*x^39+150221775*x^38+ 155903448*x^37+57663700*x^36-55834862*x^35-127637780*x^34-100327856*x^33+ 63965456*x^32+126067526*x^31+29898405*x^30-79341664*x^29-31765036*x^28+6260490* x^27+25727289*x^26+7049175*x^25-13563881*x^24+1454223*x^23+82573*x^22+957977*x^ 21+51565*x^20-692831*x^19+325486*x^18-22888*x^17+11530*x^16+204*x^15-3849*x^14-\ 5437*x^13-1289*x^12+1631*x^11+882*x^10-464*x^9+852*x^8-499*x^7-345*x^6+149*x^5+ 273*x^4-251*x^3+89*x^2-15*x+1) Theorem number, 31, Let a(n) be the, 6, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = ------------------------------- / 6 5 4 3 2 ----- -x - x - x - x - x - x + 1 n = 0 and let b(n) be the, 6, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = ------------------------------- / 6 5 4 3 2 ----- -x - x - x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n 2 17 16 15 14 13 12 ) c(n) x = 2 x (96 x + 140 x + 30 x - 206 x - 592 x - 92 x / ----- n = 0 11 10 9 8 7 6 4 3 2 + 66 x + 101 x + 94 x + 58 x - 47 x + 4 x - 2 x - 3 x - 2 x / 6 5 4 3 2 15 + 4 x - 1) / ((64 x + 32 x + 16 x + 8 x + 4 x + 2 x - 1) (x / 14 13 12 11 10 9 8 7 6 + 9 x + 60 x - 54 x - 82 x - 32 x + 54 x + 92 x - 50 x - 4 x 5 4 3 2 - 6 x - 2 x + 4 x + 6 x - 5 x + 1)) and in Maple notation 2*x^2*(96*x^17+140*x^16+30*x^15-206*x^14-592*x^13-92*x^12+66*x^11+101*x^10+94*x ^9+58*x^8-47*x^7+4*x^6-2*x^4-3*x^3-2*x^2+4*x-1)/(64*x^6+32*x^5+16*x^4+8*x^3+4*x ^2+2*x-1)/(x^15+9*x^14+60*x^13-54*x^12-82*x^11-32*x^10+54*x^9+92*x^8-50*x^7-4*x ^6-6*x^5-2*x^4+4*x^3+6*x^2-5*x+1) Theorem number, 32, Let a(n) be the, 6, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = ------------------------------- / 6 5 4 3 2 ----- -x - x - x - x - x - x + 1 n = 0 and let b(n) be the, 7, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = ------------------------------------ / 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n 2 39 38 37 36 35 ) c(n) x = - x (34 x - 629 x + 3108 x + 4317 x - 29473 x / ----- n = 0 34 33 32 31 30 - 149542 x - 264554 x + 152107 x + 762784 x + 801758 x 29 28 27 26 25 - 189173 x - 1720282 x - 706148 x + 497080 x + 944419 x 24 23 22 21 20 19 + 520756 x - 213250 x - 445815 x - 115208 x - 5092 x + 88768 x 18 17 16 15 14 13 + 89701 x + 3484 x - 64398 x + 20404 x - 1239 x - 300 x 12 11 10 9 8 7 6 5 4 + 52 x + 84 x - 3 x + 58 x + 186 x + 11 x - 88 x - 88 x + 24 x 3 2 / 42 41 40 + 94 x - 72 x + 20 x - 2) / (127 x - 3877 x + 47082 x / 39 38 37 36 35 - 215024 x - 3340 x + 1446898 x + 2472812 x - 1860737 x 34 33 32 31 30 - 8697170 x - 3661853 x + 8231002 x + 13099110 x + 1736768 x 29 28 27 26 25 - 15069924 x - 5063267 x + 2795706 x + 5530448 x + 2373381 x 24 23 22 21 20 - 1729118 x - 1805222 x - 61022 x + 56651 x + 397934 x 19 18 17 16 15 14 + 218138 x - 134932 x - 165525 x + 117458 x - 26366 x - 383 x 13 12 11 10 9 8 7 6 + 628 x + 322 x - 106 x + 350 x + 351 x - 266 x - 229 x - 28 x 5 4 3 2 + 152 x + 64 x - 136 x + 64 x - 13 x + 1) and in Maple notation -x^2*(34*x^39-629*x^38+3108*x^37+4317*x^36-29473*x^35-149542*x^34-264554*x^33+ 152107*x^32+762784*x^31+801758*x^30-189173*x^29-1720282*x^28-706148*x^27+497080 *x^26+944419*x^25+520756*x^24-213250*x^23-445815*x^22-115208*x^21-5092*x^20+ 88768*x^19+89701*x^18+3484*x^17-64398*x^16+20404*x^15-1239*x^14-300*x^13+52*x^ 12+84*x^11-3*x^10+58*x^9+186*x^8+11*x^7-88*x^6-88*x^5+24*x^4+94*x^3-72*x^2+20*x -2)/(127*x^42-3877*x^41+47082*x^40-215024*x^39-3340*x^38+1446898*x^37+2472812*x ^36-1860737*x^35-8697170*x^34-3661853*x^33+8231002*x^32+13099110*x^31+1736768*x ^30-15069924*x^29-5063267*x^28+2795706*x^27+5530448*x^26+2373381*x^25-1729118*x ^24-1805222*x^23-61022*x^22+56651*x^21+397934*x^20+218138*x^19-134932*x^18-\ 165525*x^17+117458*x^16-26366*x^15-383*x^14+628*x^13+322*x^12-106*x^11+350*x^10 +351*x^9-266*x^8-229*x^7-28*x^6+152*x^5+64*x^4-136*x^3+64*x^2-13*x+1) Theorem number, 33, Let a(n) be the, 6, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = ------------------------------- / 6 5 4 3 2 ----- -x - x - x - x - x - x + 1 n = 0 and let b(n) be the, 8, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = ----------------------------------------- / 8 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n 2 45 44 43 42 41 ) c(n) x = - x (55 x + 75 x + 590 x - 33912 x + 22680 x / ----- n = 0 40 39 38 37 36 + 356311 x + 122824 x - 1414115 x - 2850686 x - 2970787 x 35 34 33 32 31 + 5977058 x + 10516164 x + 3264386 x - 9690634 x - 13664822 x 30 29 28 27 26 + 777463 x + 7093604 x + 5264661 x + 1313534 x - 1931075 x 25 24 23 22 21 - 2700229 x - 1268709 x + 1299571 x + 455880 x + 288149 x 20 19 18 17 16 15 - 60643 x - 246675 x - 97386 x + 157887 x - 44272 x + 850 x 14 13 12 11 10 9 8 7 + 870 x + 722 x + 410 x + 94 x - 70 x + 122 x - 307 x - 121 x 6 5 4 3 2 / 48 + 64 x + 152 x + 16 x - 148 x + 90 x - 22 x + 2) / (2987 x / 47 46 45 44 43 - 29521 x + 4914 x + 429226 x + 612607 x - 4684701 x 42 41 40 39 38 - 5941299 x + 16467160 x + 33753565 x + 1797181 x - 57708743 x 37 36 35 34 - 99659261 x + 39831148 x + 127658241 x + 68011032 x 33 32 31 30 29 - 71738332 x - 108236929 x + 9904745 x + 38115906 x + 24691612 x 28 27 26 25 24 + 4861460 x - 10943002 x - 11451583 x - 2400218 x + 6751069 x 23 22 21 20 19 + 362640 x + 714557 x - 677210 x - 772759 x + 118512 x 18 17 16 15 14 13 + 476164 x - 269490 x + 44753 x + 2843 x + 1843 x + 753 x 12 11 10 9 8 7 6 5 + 71 x - 415 x + 83 x - 864 x + 159 x + 332 x + 160 x - 172 x 4 3 2 - 148 x + 188 x - 76 x + 14 x - 1) and in Maple notation -x^2*(55*x^45+75*x^44+590*x^43-33912*x^42+22680*x^41+356311*x^40+122824*x^39-\ 1414115*x^38-2850686*x^37-2970787*x^36+5977058*x^35+10516164*x^34+3264386*x^33-\ 9690634*x^32-13664822*x^31+777463*x^30+7093604*x^29+5264661*x^28+1313534*x^27-\ 1931075*x^26-2700229*x^25-1268709*x^24+1299571*x^23+455880*x^22+288149*x^21-\ 60643*x^20-246675*x^19-97386*x^18+157887*x^17-44272*x^16+850*x^15+870*x^14+722* x^13+410*x^12+94*x^11-70*x^10+122*x^9-307*x^8-121*x^7+64*x^6+152*x^5+16*x^4-148 *x^3+90*x^2-22*x+2)/(2987*x^48-29521*x^47+4914*x^46+429226*x^45+612607*x^44-\ 4684701*x^43-5941299*x^42+16467160*x^41+33753565*x^40+1797181*x^39-57708743*x^ 38-99659261*x^37+39831148*x^36+127658241*x^35+68011032*x^34-71738332*x^33-\ 108236929*x^32+9904745*x^31+38115906*x^30+24691612*x^29+4861460*x^28-10943002*x ^27-11451583*x^26-2400218*x^25+6751069*x^24+362640*x^23+714557*x^22-677210*x^21 -772759*x^20+118512*x^19+476164*x^18-269490*x^17+44753*x^16+2843*x^15+1843*x^14 +753*x^13+71*x^12-415*x^11+83*x^10-864*x^9+159*x^8+332*x^7+160*x^6-172*x^5-148* x^4+188*x^3-76*x^2+14*x-1) Theorem number, 34, Let a(n) be the, 6, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = ------------------------------- / 6 5 4 3 2 ----- -x - x - x - x - x - x + 1 n = 0 and let b(n) be the, 9, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = ---------------------------------------------- / 9 8 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n 2 51 50 49 48 47 ) c(n) x = x (177 x + 3391 x - 14517 x - 59106 x + 191066 x / ----- n = 0 46 45 44 43 42 + 1108930 x - 519916 x - 3719573 x - 9799620 x + 2164308 x 41 40 39 38 37 + 31710155 x + 48905951 x + 774026 x - 113908080 x - 77435652 x 36 35 34 33 + 45182069 x + 109789712 x + 56904587 x - 43801132 x 32 31 30 29 28 - 47173884 x - 30210695 x + 5342990 x + 17123092 x + 16283544 x 27 26 25 24 23 + 4716255 x - 8220005 x - 7876585 x + 2556453 x + 664930 x 22 21 20 19 18 + 998219 x + 315832 x - 455353 x - 375645 x + 359394 x 17 16 15 14 13 12 11 - 84389 x - 2310 x - 2043 x - 202 x + 819 x + 759 x + 62 x 10 9 8 7 6 5 4 3 + 43 x + 465 x - 360 x - 273 x - 26 x + 214 x + 94 x - 218 x 2 / 54 53 52 51 + 110 x - 24 x + 2) / (2503 x + 28231 x + 2745 x - 1467603 x / 50 49 48 47 46 + 788666 x + 15138210 x + 11101151 x - 74618092 x - 86863784 x 45 44 43 42 + 31055500 x + 389463171 x + 323837516 x - 306530822 x 41 40 39 38 - 1022040381 x - 463806769 x + 1204750235 x + 922171423 x 37 36 35 34 - 231282994 x - 820945653 x - 394873001 x + 281842335 x 33 32 31 30 + 211879617 x + 155438929 x - 39346493 x - 74738775 x 29 28 27 26 25 - 68044540 x - 4997914 x + 45144658 x + 19020388 x - 17841320 x 24 23 22 21 20 + 434288 x - 3153974 x + 433984 x + 1957804 x + 215389 x 19 18 17 16 15 14 - 1230165 x + 550838 x - 70744 x + 6634 x - 788 x - 3119 x 13 12 11 10 9 8 7 - 2200 x + 544 x - 353 x - 541 x + 1377 x + 125 x - 366 x 6 5 4 3 2 - 355 x + 149 x + 273 x - 251 x + 89 x - 15 x + 1) and in Maple notation x^2*(177*x^51+3391*x^50-14517*x^49-59106*x^48+191066*x^47+1108930*x^46-519916*x ^45-3719573*x^44-9799620*x^43+2164308*x^42+31710155*x^41+48905951*x^40+774026*x ^39-113908080*x^38-77435652*x^37+45182069*x^36+109789712*x^35+56904587*x^34-\ 43801132*x^33-47173884*x^32-30210695*x^31+5342990*x^30+17123092*x^29+16283544*x ^28+4716255*x^27-8220005*x^26-7876585*x^25+2556453*x^24+664930*x^23+998219*x^22 +315832*x^21-455353*x^20-375645*x^19+359394*x^18-84389*x^17-2310*x^16-2043*x^15 -202*x^14+819*x^13+759*x^12+62*x^11+43*x^10+465*x^9-360*x^8-273*x^7-26*x^6+214* x^5+94*x^4-218*x^3+110*x^2-24*x+2)/(2503*x^54+28231*x^53+2745*x^52-1467603*x^51 +788666*x^50+15138210*x^49+11101151*x^48-74618092*x^47-86863784*x^46+31055500*x ^45+389463171*x^44+323837516*x^43-306530822*x^42-1022040381*x^41-463806769*x^40 +1204750235*x^39+922171423*x^38-231282994*x^37-820945653*x^36-394873001*x^35+ 281842335*x^34+211879617*x^33+155438929*x^32-39346493*x^31-74738775*x^30-\ 68044540*x^29-4997914*x^28+45144658*x^27+19020388*x^26-17841320*x^25+434288*x^ 24-3153974*x^23+433984*x^22+1957804*x^21+215389*x^20-1230165*x^19+550838*x^18-\ 70744*x^17+6634*x^16-788*x^15-3119*x^14-2200*x^13+544*x^12-353*x^11-541*x^10+ 1377*x^9+125*x^8-366*x^7-355*x^6+149*x^5+273*x^4-251*x^3+89*x^2-15*x+1) Theorem number, 35, Let a(n) be the, 6, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = ------------------------------- / 6 5 4 3 2 ----- -x - x - x - x - x - x + 1 n = 0 and let b(n) be the, 10, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = ---------------------------------------------------- / 10 9 8 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n 2 57 56 55 54 ) c(n) x = - x (105 x - 2973 x - 15404 x - 164429 x / ----- n = 0 53 52 51 50 49 + 615025 x + 1063207 x - 2180059 x - 18367271 x - 9373541 x 48 47 46 45 + 28946723 x + 113672092 x + 187423730 x - 211266871 x 44 43 42 41 - 639809503 x - 347200443 x + 620207019 x + 1192885814 x 40 39 38 37 + 38859584 x - 805241988 x - 692878831 x - 12618428 x 36 35 34 33 + 378547160 x + 249007741 x + 165303682 x - 72355013 x 32 31 30 29 28 - 183126665 x - 69320307 x + 9085602 x + 59269515 x + 41695014 x 27 26 25 24 23 - 8556753 x - 28288277 x + 4910468 x - 445100 x + 1958380 x 22 21 20 19 18 + 1433276 x - 593157 x - 1087100 x + 761595 x - 134924 x 17 16 15 14 13 12 11 + 1161 x - 2924 x - 4310 x - 1752 x + 268 x + 648 x + 35 x 10 9 8 7 6 5 4 3 - 71 x + 1034 x - 261 x - 423 x - 204 x + 250 x + 224 x - 306 x 2 / 60 59 58 57 + 132 x - 26 x + 2) / (4591 x - 86509 x + 414147 x + 1723154 x / 56 55 54 53 - 1833403 x - 36966284 x + 19810131 x + 209767701 x 52 51 50 49 + 272874185 x - 630411506 x - 1197328331 x - 1014038784 x 48 47 46 45 + 2203954774 x + 6585234152 x + 239805255 x - 9388700106 x 44 43 42 41 - 8030040267 x + 4630610180 x + 12293286218 x + 1062480488 x 40 39 38 37 - 5370247087 x - 4581598924 x + 2112076 x + 1876504889 x 36 35 34 33 + 1022608632 x + 1040065501 x - 488883470 x - 871600025 x 32 31 30 29 - 179205532 x + 95770485 x + 279844552 x + 102126276 x 28 27 26 25 24 - 98876824 x - 82689649 x + 46268896 x - 6784153 x + 8331874 x 23 22 21 20 19 + 1640199 x - 4130711 x - 1588141 x + 2865035 x - 1018061 x 18 17 16 15 14 13 + 138034 x - 9915 x - 14822 x - 2585 x + 3632 x + 1456 x 12 11 10 9 8 7 6 - 107 x + 662 x + 1750 x - 1802 x - 586 x + 256 x + 601 x 5 4 3 2 - 52 x - 449 x + 326 x - 103 x + 16 x - 1) and in Maple notation -x^2*(105*x^57-2973*x^56-15404*x^55-164429*x^54+615025*x^53+1063207*x^52-\ 2180059*x^51-18367271*x^50-9373541*x^49+28946723*x^48+113672092*x^47+187423730* x^46-211266871*x^45-639809503*x^44-347200443*x^43+620207019*x^42+1192885814*x^ 41+38859584*x^40-805241988*x^39-692878831*x^38-12618428*x^37+378547160*x^36+ 249007741*x^35+165303682*x^34-72355013*x^33-183126665*x^32-69320307*x^31+ 9085602*x^30+59269515*x^29+41695014*x^28-8556753*x^27-28288277*x^26+4910468*x^ 25-445100*x^24+1958380*x^23+1433276*x^22-593157*x^21-1087100*x^20+761595*x^19-\ 134924*x^18+1161*x^17-2924*x^16-4310*x^15-1752*x^14+268*x^13+648*x^12+35*x^11-\ 71*x^10+1034*x^9-261*x^8-423*x^7-204*x^6+250*x^5+224*x^4-306*x^3+132*x^2-26*x+2 )/(4591*x^60-86509*x^59+414147*x^58+1723154*x^57-1833403*x^56-36966284*x^55+ 19810131*x^54+209767701*x^53+272874185*x^52-630411506*x^51-1197328331*x^50-\ 1014038784*x^49+2203954774*x^48+6585234152*x^47+239805255*x^46-9388700106*x^45-\ 8030040267*x^44+4630610180*x^43+12293286218*x^42+1062480488*x^41-5370247087*x^ 40-4581598924*x^39+2112076*x^38+1876504889*x^37+1022608632*x^36+1040065501*x^35 -488883470*x^34-871600025*x^33-179205532*x^32+95770485*x^31+279844552*x^30+ 102126276*x^29-98876824*x^28-82689649*x^27+46268896*x^26-6784153*x^25+8331874*x ^24+1640199*x^23-4130711*x^22-1588141*x^21+2865035*x^20-1018061*x^19+138034*x^ 18-9915*x^17-14822*x^16-2585*x^15+3632*x^14+1456*x^13-107*x^12+662*x^11+1750*x^ 10-1802*x^9-586*x^8+256*x^7+601*x^6-52*x^5-449*x^4+326*x^3-103*x^2+16*x-1) Theorem number, 36, Let a(n) be the, 7, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = ------------------------------------ / 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x + 1 n = 0 and let b(n) be the, 7, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = ------------------------------------ / 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n 2 21 20 19 18 17 ) c(n) x = 2 x (4168 x + 3828 x + 148 x - 3480 x - 4956 x / ----- n = 0 16 15 14 13 12 11 10 - 3528 x + 2548 x + 1108 x + 500 x + 192 x - 8 x - 162 x 9 8 7 6 5 3 2 / - 210 x + 98 x - 14 x - 8 x - 4 x + 4 x + 5 x - 5 x + 1) / ( / 7 6 5 4 3 2 21 20 (128 x + 64 x + 32 x + 16 x + 8 x + 4 x + 2 x - 1) (8 x - 24 x 19 18 17 16 15 14 13 - 48 x + 56 x + 368 x + 504 x - 896 x - 396 x + 56 x 12 11 10 9 8 7 6 5 4 + 332 x + 304 x - 8 x - 312 x + 84 x - 14 x + 2 x + 10 x + 8 x 3 2 - 2 x - 10 x + 6 x - 1)) and in Maple notation 2*x^2*(4168*x^21+3828*x^20+148*x^19-3480*x^18-4956*x^17-3528*x^16+2548*x^15+ 1108*x^14+500*x^13+192*x^12-8*x^11-162*x^10-210*x^9+98*x^8-14*x^7-8*x^6-4*x^5+4 *x^3+5*x^2-5*x+1)/(128*x^7+64*x^6+32*x^5+16*x^4+8*x^3+4*x^2+2*x-1)/(8*x^21-24*x ^20-48*x^19+56*x^18+368*x^17+504*x^16-896*x^15-396*x^14+56*x^13+332*x^12+304*x^ 11-8*x^10-312*x^9+84*x^8-14*x^7+2*x^6+10*x^5+8*x^4-2*x^3-10*x^2+6*x-1) Theorem number, 37, Let a(n) be the, 7, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = ------------------------------------ / 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x + 1 n = 0 and let b(n) be the, 8, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = ----------------------------------------- / 8 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n 2 53 52 51 50 49 ) c(n) x = x (278 x + 5479 x + 23543 x - 40524 x - 266887 x / ----- n = 0 48 47 46 45 44 - 225067 x + 1685113 x + 7967710 x - 10277148 x - 30939701 x 43 42 41 40 39 - 5405632 x + 70913081 x + 114924127 x + 7231442 x - 185361485 x 38 37 36 35 - 167657318 x + 4713748 x + 164897747 x + 174446849 x 34 33 32 31 30 + 15859726 x - 151861216 x - 63133087 x + 1522541 x + 33623308 x 29 28 27 26 25 + 29562748 x + 6799272 x - 10026827 x - 7966862 x - 243984 x 24 23 22 21 20 - 813740 x + 1073399 x + 1419620 x + 457085 x - 641206 x 19 18 17 16 15 14 - 546930 x + 502142 x - 111129 x + 407 x - 149 x - 403 x 13 12 11 10 9 8 7 6 - 473 x - 151 x + 563 x + 691 x - 17 x - 248 x - 282 x - 26 x 5 4 3 2 / 56 55 + 214 x + 94 x - 218 x + 110 x - 24 x + 2) / (9851 x + 74724 x / 54 53 52 51 50 + 215562 x - 640342 x - 4622576 x - 1046656 x + 28953424 x 49 48 47 46 + 60386039 x - 43137327 x - 473627500 x + 14855936 x 45 44 43 42 + 944083730 x + 944474858 x - 614510888 x - 2104685345 x 41 40 39 38 - 938793106 x + 1620557107 x + 1791412482 x + 380768378 x 37 36 35 34 - 1137250018 x - 1301742186 x - 40380329 x + 926519775 x 33 32 31 30 + 183090113 x - 18095641 x - 155253756 x - 115339436 x 29 28 27 26 25 - 676852 x + 51054409 x + 17231652 x - 5240474 x + 3991452 x 24 23 22 21 20 - 5841031 x - 4094368 x + 669398 x + 2668859 x + 304767 x 19 18 17 16 15 14 - 1647678 x + 728088 x - 99339 x + 701 x + 692 x + 1253 x 13 12 11 10 9 8 7 - 364 x - 2507 x - 1052 x + 1043 x + 726 x + 243 x - 375 x 6 5 4 3 2 - 355 x + 149 x + 273 x - 251 x + 89 x - 15 x + 1) and in Maple notation x^2*(278*x^53+5479*x^52+23543*x^51-40524*x^50-266887*x^49-225067*x^48+1685113*x ^47+7967710*x^46-10277148*x^45-30939701*x^44-5405632*x^43+70913081*x^42+ 114924127*x^41+7231442*x^40-185361485*x^39-167657318*x^38+4713748*x^37+ 164897747*x^36+174446849*x^35+15859726*x^34-151861216*x^33-63133087*x^32+ 1522541*x^31+33623308*x^30+29562748*x^29+6799272*x^28-10026827*x^27-7966862*x^ 26-243984*x^25-813740*x^24+1073399*x^23+1419620*x^22+457085*x^21-641206*x^20-\ 546930*x^19+502142*x^18-111129*x^17+407*x^16-149*x^15-403*x^14-473*x^13-151*x^ 12+563*x^11+691*x^10-17*x^9-248*x^8-282*x^7-26*x^6+214*x^5+94*x^4-218*x^3+110*x ^2-24*x+2)/(9851*x^56+74724*x^55+215562*x^54-640342*x^53-4622576*x^52-1046656*x ^51+28953424*x^50+60386039*x^49-43137327*x^48-473627500*x^47+14855936*x^46+ 944083730*x^45+944474858*x^44-614510888*x^43-2104685345*x^42-938793106*x^41+ 1620557107*x^40+1791412482*x^39+380768378*x^38-1137250018*x^37-1301742186*x^36-\ 40380329*x^35+926519775*x^34+183090113*x^33-18095641*x^32-155253756*x^31-\ 115339436*x^30-676852*x^29+51054409*x^28+17231652*x^27-5240474*x^26+3991452*x^ 25-5841031*x^24-4094368*x^23+669398*x^22+2668859*x^21+304767*x^20-1647678*x^19+ 728088*x^18-99339*x^17+701*x^16+692*x^15+1253*x^14-364*x^13-2507*x^12-1052*x^11 +1043*x^10+726*x^9+243*x^8-375*x^7-355*x^6+149*x^5+273*x^4-251*x^3+89*x^2-15*x+ 1) Theorem number, 38, Let a(n) be the, 7, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = ------------------------------------ / 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x + 1 n = 0 and let b(n) be the, 9, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = ---------------------------------------------- / 9 8 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n 2 60 59 58 57 ) c(n) x = - x (59 x + 1023 x - 18868 x - 306332 x / ----- n = 0 56 55 54 53 52 - 542381 x - 3209387 x + 6367281 x + 26140613 x + 3991920 x 51 50 49 48 - 125986548 x - 237999635 x + 282282771 x + 856354792 x 47 46 45 44 + 208260278 x - 1161606751 x - 1671581147 x - 463120190 x 43 42 41 40 + 1585029690 x + 2905847409 x + 317013605 x - 1887204403 x 39 38 37 36 - 2004471651 x - 415486951 x + 1174905999 x + 1203000297 x 35 34 33 32 - 117415429 x - 298862530 x - 238250264 x - 115388709 x 31 30 29 28 27 + 17088379 x + 78074754 x + 58610558 x + 5725343 x - 31455839 x 26 25 24 23 22 + 1277851 x - 5953127 x - 264052 x + 3243440 x + 2392971 x 21 20 19 18 17 16 - 916437 x - 1810360 x + 1184526 x - 212549 x + 1713 x - 267 x 15 14 13 12 11 10 9 - 931 x - 1003 x - 849 x - 543 x - 203 x + 1238 x + 336 x 8 7 6 5 4 3 2 - 116 x - 434 x - 204 x + 250 x + 224 x - 306 x + 132 x - 26 x + 2) / 63 62 61 60 59 / (778 x - 19594 x - 45702 x + 1912636 x + 13265086 x / 58 57 56 55 + 3559378 x - 11298254 x - 381938243 x - 33236260 x 54 53 52 51 + 1578002288 x + 2285966392 x - 2983324876 x - 10582954860 x 50 49 48 47 + 297156971 x + 17855787530 x + 13330866521 x - 8561105720 x 46 45 44 43 - 23708472740 x - 14379374432 x + 10825730885 x + 28525490030 x 42 41 40 39 + 4888571772 x - 12029379462 x - 13783061155 x - 2637558804 x 38 37 36 35 + 7424276355 x + 5635134266 x - 1765753577 x - 914923960 x 34 33 32 31 - 820205535 x - 424070962 x + 150894808 x + 326858514 x 30 29 28 27 + 159291294 x - 49732878 x - 108772418 x + 34445194 x 26 25 24 23 22 - 26048001 x + 8193988 x + 12584374 x + 2422592 x - 6337142 x 21 20 19 18 17 - 2485608 x + 4274753 x - 1526646 x + 192525 x - 1936 x 16 15 14 13 12 11 10 - 3032 x - 3402 x - 2377 x - 64 x + 1078 x + 3380 x - 729 x 9 8 7 6 5 4 3 2 - 946 x - 728 x + 266 x + 601 x - 52 x - 449 x + 326 x - 103 x + 16 x - 1) and in Maple notation -x^2*(59*x^60+1023*x^59-18868*x^58-306332*x^57-542381*x^56-3209387*x^55+6367281 *x^54+26140613*x^53+3991920*x^52-125986548*x^51-237999635*x^50+282282771*x^49+ 856354792*x^48+208260278*x^47-1161606751*x^46-1671581147*x^45-463120190*x^44+ 1585029690*x^43+2905847409*x^42+317013605*x^41-1887204403*x^40-2004471651*x^39-\ 415486951*x^38+1174905999*x^37+1203000297*x^36-117415429*x^35-298862530*x^34-\ 238250264*x^33-115388709*x^32+17088379*x^31+78074754*x^30+58610558*x^29+5725343 *x^28-31455839*x^27+1277851*x^26-5953127*x^25-264052*x^24+3243440*x^23+2392971* x^22-916437*x^21-1810360*x^20+1184526*x^19-212549*x^18+1713*x^17-267*x^16-931*x ^15-1003*x^14-849*x^13-543*x^12-203*x^11+1238*x^10+336*x^9-116*x^8-434*x^7-204* x^6+250*x^5+224*x^4-306*x^3+132*x^2-26*x+2)/(778*x^63-19594*x^62-45702*x^61+ 1912636*x^60+13265086*x^59+3559378*x^58-11298254*x^57-381938243*x^56-33236260*x ^55+1578002288*x^54+2285966392*x^53-2983324876*x^52-10582954860*x^51+297156971* x^50+17855787530*x^49+13330866521*x^48-8561105720*x^47-23708472740*x^46-\ 14379374432*x^45+10825730885*x^44+28525490030*x^43+4888571772*x^42-12029379462* x^41-13783061155*x^40-2637558804*x^39+7424276355*x^38+5635134266*x^37-\ 1765753577*x^36-914923960*x^35-820205535*x^34-424070962*x^33+150894808*x^32+ 326858514*x^31+159291294*x^30-49732878*x^29-108772418*x^28+34445194*x^27-\ 26048001*x^26+8193988*x^25+12584374*x^24+2422592*x^23-6337142*x^22-2485608*x^21 +4274753*x^20-1526646*x^19+192525*x^18-1936*x^17-3032*x^16-3402*x^15-2377*x^14-\ 64*x^13+1078*x^12+3380*x^11-729*x^10-946*x^9-728*x^8+266*x^7+601*x^6-52*x^5-449 *x^4+326*x^3-103*x^2+16*x-1) Theorem number, 39, Let a(n) be the, 7, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = ------------------------------------ / 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x + 1 n = 0 and let b(n) be the, 10, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = ---------------------------------------------------- / 10 9 8 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n 2 67 66 65 64 ) c(n) x = - x (312 x - 12863 x + 38336 x - 247187 x / ----- n = 0 63 62 61 60 59 + 394610 x - 34837 x - 3497147 x - 65658886 x - 239426706 x 58 57 56 55 + 392791908 x + 1323497713 x - 264376081 x - 4218236146 x 54 53 52 51 - 4399451816 x + 5866924230 x + 13049810900 x + 8146361371 x 50 49 48 47 - 14171235436 x - 25272529413 x - 11971671679 x + 15536912267 x 46 45 44 43 + 34190343695 x + 17244195504 x - 23504560890 x - 23783413599 x 42 41 40 39 - 5716237293 x + 9672033343 x + 11593598273 x + 2870137135 x 38 37 36 35 - 4199372668 x - 1950146436 x - 1382858018 x + 3946396 x 34 33 32 31 + 419168657 x + 584627815 x + 315073210 x - 72911632 x 30 29 28 27 - 256659127 x - 80142289 x + 93870974 x - 18508936 x 26 25 24 23 22 + 17461970 x + 7772118 x - 4470016 x - 7420465 x + 133866 x 21 20 19 18 17 16 + 5046174 x - 2526472 x + 388717 x - 10605 x - 3986 x - 818 x 15 14 13 12 11 10 9 + 581 x + 399 x + 308 x + 1778 x + 1435 x - 1486 x - 790 x 8 7 6 5 4 3 2 - 248 x + 506 x + 480 x - 224 x - 422 x + 414 x - 156 x + 28 x - 2) / 70 69 68 67 66 / (2719 x - 18941 x + 1027898 x - 3034411 x + 3134173 x / 65 64 63 62 - 47525132 x + 63485993 x + 527107608 x + 2191595478 x 61 60 59 58 - 1903240376 x - 18849346769 x + 2081980948 x + 58549658777 x 57 56 55 54 + 40051122044 x - 87160932245 x - 164463353242 x + 14384655027 x 53 52 51 50 + 209491354552 x + 231969656566 x - 75556492660 x - 287912617960 x 49 48 47 46 - 218260550592 x + 73162515774 x + 313255195616 x + 159755822822 x 45 44 43 42 - 170812553297 x - 140522217166 x - 30790225157 x + 55768386653 x 41 40 39 38 + 57213945608 x + 4369894899 x - 21197035879 x - 2795163511 x 37 36 35 34 - 5893652169 x + 510769028 x + 1481524026 x + 2349786496 x 33 32 31 30 + 829898585 x - 714496006 x - 923064622 x + 79203762 x 29 28 27 26 + 343360973 x - 150626715 x + 89044493 x + 1377175 x 25 24 23 22 21 - 28610785 x - 14874419 x + 11968391 x + 9636671 x - 9931670 x 20 19 18 17 16 15 + 3050452 x - 380062 x - 9848 x - 1892 x + 3730 x + 2552 x 14 13 12 11 10 9 8 + 1570 x + 5318 x - 13 x - 5315 x - 260 x + 807 x + 1333 x 7 6 5 4 3 2 + 72 x - 864 x - 159 x + 687 x - 414 x + 118 x - 17 x + 1) and in Maple notation -x^2*(312*x^67-12863*x^66+38336*x^65-247187*x^64+394610*x^63-34837*x^62-3497147 *x^61-65658886*x^60-239426706*x^59+392791908*x^58+1323497713*x^57-264376081*x^ 56-4218236146*x^55-4399451816*x^54+5866924230*x^53+13049810900*x^52+8146361371* x^51-14171235436*x^50-25272529413*x^49-11971671679*x^48+15536912267*x^47+ 34190343695*x^46+17244195504*x^45-23504560890*x^44-23783413599*x^43-5716237293* x^42+9672033343*x^41+11593598273*x^40+2870137135*x^39-4199372668*x^38-\ 1950146436*x^37-1382858018*x^36+3946396*x^35+419168657*x^34+584627815*x^33+ 315073210*x^32-72911632*x^31-256659127*x^30-80142289*x^29+93870974*x^28-\ 18508936*x^27+17461970*x^26+7772118*x^25-4470016*x^24-7420465*x^23+133866*x^22+ 5046174*x^21-2526472*x^20+388717*x^19-10605*x^18-3986*x^17-818*x^16+581*x^15+ 399*x^14+308*x^13+1778*x^12+1435*x^11-1486*x^10-790*x^9-248*x^8+506*x^7+480*x^6 -224*x^5-422*x^4+414*x^3-156*x^2+28*x-2)/(2719*x^70-18941*x^69+1027898*x^68-\ 3034411*x^67+3134173*x^66-47525132*x^65+63485993*x^64+527107608*x^63+2191595478 *x^62-1903240376*x^61-18849346769*x^60+2081980948*x^59+58549658777*x^58+ 40051122044*x^57-87160932245*x^56-164463353242*x^55+14384655027*x^54+ 209491354552*x^53+231969656566*x^52-75556492660*x^51-287912617960*x^50-\ 218260550592*x^49+73162515774*x^48+313255195616*x^47+159755822822*x^46-\ 170812553297*x^45-140522217166*x^44-30790225157*x^43+55768386653*x^42+ 57213945608*x^41+4369894899*x^40-21197035879*x^39-2795163511*x^38-5893652169*x^ 37+510769028*x^36+1481524026*x^35+2349786496*x^34+829898585*x^33-714496006*x^32 -923064622*x^31+79203762*x^30+343360973*x^29-150626715*x^28+89044493*x^27+ 1377175*x^26-28610785*x^25-14874419*x^24+11968391*x^23+9636671*x^22-9931670*x^ 21+3050452*x^20-380062*x^19-9848*x^18-1892*x^17+3730*x^16+2552*x^15+1570*x^14+ 5318*x^13-13*x^12-5315*x^11-260*x^10+807*x^9+1333*x^8+72*x^7-864*x^6-159*x^5+ 687*x^4-414*x^3+118*x^2-17*x+1) Theorem number, 40, Let a(n) be the, 8, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = ----------------------------------------- / 8 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x - x + 1 n = 0 and let b(n) be the, 8, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = ----------------------------------------- / 8 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n 2 31 30 29 28 27 ) c(n) x = 2 x (4096 x + 5248 x - 4700 x - 20034 x - 18040 x / ----- n = 0 26 25 24 23 22 21 + 24904 x + 140700 x + 51652 x - 38292 x - 72382 x - 57717 x 20 19 18 17 16 15 - 19787 x + 18672 x + 37602 x - 3116 x - 842 x - 572 x 14 13 12 11 10 9 8 7 - 689 x - 718 x - 469 x + 88 x + 588 x - 240 x + 6 x + 7 x 6 5 4 3 2 / + 9 x + 9 x + 5 x - 3 x - 9 x + 6 x - 1) / ( / 8 7 6 5 4 3 2 28 (256 x + 128 x + 64 x + 32 x + 16 x + 8 x + 4 x + 2 x - 1) (x 27 26 25 24 23 22 21 + 12 x + 104 x + 846 x - 1358 x - 2120 x + 354 x + 4513 x 20 19 18 17 16 15 + 5355 x - 1380 x - 10980 x + 1408 x + 2024 x + 1936 x 14 13 12 11 10 9 8 + 905 x - 410 x - 1063 x - 462 x + 795 x - 210 x + 36 x 7 6 5 4 3 2 + 27 x + 9 x - 9 x - 15 x - 3 x + 15 x - 7 x + 1)) and in Maple notation 2*x^2*(4096*x^31+5248*x^30-4700*x^29-20034*x^28-18040*x^27+24904*x^26+140700*x^ 25+51652*x^24-38292*x^23-72382*x^22-57717*x^21-19787*x^20+18672*x^19+37602*x^18 -3116*x^17-842*x^16-572*x^15-689*x^14-718*x^13-469*x^12+88*x^11+588*x^10-240*x^ 9+6*x^8+7*x^7+9*x^6+9*x^5+5*x^4-3*x^3-9*x^2+6*x-1)/(256*x^8+128*x^7+64*x^6+32*x ^5+16*x^4+8*x^3+4*x^2+2*x-1)/(x^28+12*x^27+104*x^26+846*x^25-1358*x^24-2120*x^ 23+354*x^22+4513*x^21+5355*x^20-1380*x^19-10980*x^18+1408*x^17+2024*x^16+1936*x ^15+905*x^14-410*x^13-1063*x^12-462*x^11+795*x^10-210*x^9+36*x^8+27*x^7+9*x^6-9 *x^5-15*x^4-3*x^3+15*x^2-7*x+1) Theorem number, 41, Let a(n) be the, 8, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = ----------------------------------------- / 8 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x - x + 1 n = 0 and let b(n) be the, 9, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = ---------------------------------------------- / 9 8 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n 2 69 68 67 66 ) c(n) x = - x (138 x - 5665 x + 86770 x - 537730 x / ----- n = 0 65 64 63 62 61 + 258606 x + 7145763 x + 3874435 x - 78436321 x - 362896235 x 60 59 58 57 - 470095077 x + 1302792710 x + 3518891188 x + 1060705853 x 56 55 54 53 - 8066189567 x - 15045573287 x - 2345517744 x + 37247588555 x 52 51 50 49 + 30800150765 x - 18444716458 x - 56783590004 x - 39182311778 x 48 47 46 45 + 22549684636 x + 64813628049 x + 36531807248 x - 23589587775 x 44 43 42 41 - 38076165461 x - 22582495538 x + 3202628094 x + 19124979329 x 40 39 38 37 + 15334169837 x - 871938544 x - 9969433126 x - 1690597979 x 36 35 34 33 - 353466408 x + 818981079 x + 964051311 x + 427279701 x 32 31 30 29 - 121589540 x - 247149753 x - 48741264 x + 30706279 x 28 27 26 25 24 - 40040244 x + 15489242 x + 22132034 x + 10147469 x - 5967496 x 23 22 21 20 19 - 10037400 x + 14109 x + 6735191 x - 3312011 x + 517326 x 18 17 16 15 14 13 - 4104 x - 4850 x - 4574 x - 2601 x + 619 x + 2990 x 12 11 10 9 8 7 6 + 2016 x - 221 x - 687 x - 939 x - 238 x + 506 x + 480 x 5 4 3 2 / 72 71 - 224 x - 422 x + 414 x - 156 x + 28 x - 2) / (529 x - 30130 x / 70 69 68 67 66 + 711801 x - 8665020 x + 52021678 x - 84804302 x - 391228025 x 65 64 63 62 + 491552659 x + 4268565764 x + 5862462027 x - 14405143144 x 61 60 59 58 - 45779177898 x + 886440289 x + 117004261215 x + 137806668991 x 57 56 55 54 - 85068898403 x - 390366419604 x - 262279175759 x + 511551016266 x 53 52 51 50 + 581217022847 x + 8360427699 x - 579174695230 x - 542877031948 x 49 48 47 46 + 73707442099 x + 532324701492 x + 305247955523 x - 133238699790 x 45 44 43 42 - 213905641080 x - 142480237018 x + 12949586848 x + 106744891065 x 41 40 39 38 + 70187260353 x - 21876175659 x - 43901515927 x + 5140094228 x 37 36 35 34 - 2675087110 x + 3270578677 x + 3443325425 x + 852059454 x 33 32 31 30 - 982286650 x - 716945270 x + 168586789 x + 60247157 x 29 28 27 26 - 182840107 x + 115714147 x + 65832827 x + 1670610 x 25 24 23 22 21 - 37163444 x - 19326842 x + 15394662 x + 12444933 x - 12768280 x 20 19 18 17 16 15 + 3927519 x - 445419 x - 12277 x - 13736 x - 5740 x + 6817 x 14 13 12 11 10 9 8 + 10415 x + 1747 x - 3508 x - 2210 x - 1248 x + 958 x + 1323 x 7 6 5 4 3 2 + 72 x - 864 x - 159 x + 687 x - 414 x + 118 x - 17 x + 1) and in Maple notation -x^2*(138*x^69-5665*x^68+86770*x^67-537730*x^66+258606*x^65+7145763*x^64+ 3874435*x^63-78436321*x^62-362896235*x^61-470095077*x^60+1302792710*x^59+ 3518891188*x^58+1060705853*x^57-8066189567*x^56-15045573287*x^55-2345517744*x^ 54+37247588555*x^53+30800150765*x^52-18444716458*x^51-56783590004*x^50-\ 39182311778*x^49+22549684636*x^48+64813628049*x^47+36531807248*x^46-23589587775 *x^45-38076165461*x^44-22582495538*x^43+3202628094*x^42+19124979329*x^41+ 15334169837*x^40-871938544*x^39-9969433126*x^38-1690597979*x^37-353466408*x^36+ 818981079*x^35+964051311*x^34+427279701*x^33-121589540*x^32-247149753*x^31-\ 48741264*x^30+30706279*x^29-40040244*x^28+15489242*x^27+22132034*x^26+10147469* x^25-5967496*x^24-10037400*x^23+14109*x^22+6735191*x^21-3312011*x^20+517326*x^ 19-4104*x^18-4850*x^17-4574*x^16-2601*x^15+619*x^14+2990*x^13+2016*x^12-221*x^ 11-687*x^10-939*x^9-238*x^8+506*x^7+480*x^6-224*x^5-422*x^4+414*x^3-156*x^2+28* x-2)/(529*x^72-30130*x^71+711801*x^70-8665020*x^69+52021678*x^68-84804302*x^67-\ 391228025*x^66+491552659*x^65+4268565764*x^64+5862462027*x^63-14405143144*x^62-\ 45779177898*x^61+886440289*x^60+117004261215*x^59+137806668991*x^58-85068898403 *x^57-390366419604*x^56-262279175759*x^55+511551016266*x^54+581217022847*x^53+ 8360427699*x^52-579174695230*x^51-542877031948*x^50+73707442099*x^49+ 532324701492*x^48+305247955523*x^47-133238699790*x^46-213905641080*x^45-\ 142480237018*x^44+12949586848*x^43+106744891065*x^42+70187260353*x^41-\ 21876175659*x^40-43901515927*x^39+5140094228*x^38-2675087110*x^37+3270578677*x^ 36+3443325425*x^35+852059454*x^34-982286650*x^33-716945270*x^32+168586789*x^31+ 60247157*x^30-182840107*x^29+115714147*x^28+65832827*x^27+1670610*x^26-37163444 *x^25-19326842*x^24+15394662*x^23+12444933*x^22-12768280*x^21+3927519*x^20-\ 445419*x^19-12277*x^18-13736*x^17-5740*x^16+6817*x^15+10415*x^14+1747*x^13-3508 *x^12-2210*x^11-1248*x^10+958*x^9+1323*x^8+72*x^7-864*x^6-159*x^5+687*x^4-414*x ^3+118*x^2-17*x+1) Theorem number, 42, Let a(n) be the, 8, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = ----------------------------------------- / 8 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x - x + 1 n = 0 and let b(n) be the, 10, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = ---------------------------------------------------- / 10 9 8 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n 2 77 76 75 74 ) c(n) x = - x (917 x - 11376 x + 32295 x - 214814 x / ----- n = 0 73 72 71 70 69 - 127021 x + 24159368 x - 47647205 x - 356797187 x + 329761349 x 68 67 66 65 + 2507526014 x + 838820680 x - 12473599399 x - 24463724767 x 64 63 62 61 - 19665809142 x + 96843153613 x + 165174024880 x - 27632854558 x 60 59 58 - 390157991950 x - 430075280392 x + 226409252434 x 57 56 55 + 1011925317720 x + 481444474683 x - 672419513060 x 54 53 52 - 1033886918259 x - 431399887393 x + 470927426947 x 51 50 49 + 876216115226 x + 501321702377 x - 240935777781 x 48 47 46 - 688447982122 x - 240990587840 x + 108174861719 x 45 44 43 42 + 241144623411 x + 155481500342 x - 12736907544 x - 101054185746 x 41 40 39 38 - 53845491245 x + 24984998494 x + 4886316375 x + 6474065457 x 37 36 35 34 + 4697325822 x + 671266433 x - 1709454186 x - 1761379266 x 33 32 31 30 - 560102825 x + 399498037 x + 463334997 x - 257947382 x 29 28 27 26 25 + 181888069 x + 10415147 x - 45444863 x - 42942779 x - 778082 x 24 23 22 21 20 + 26752604 x + 7645341 x - 18420145 x + 7370539 x - 1021359 x 19 18 17 16 15 14 - 7059 x - 1816 x + 2994 x + 5573 x + 5293 x + 2592 x 13 12 11 10 9 8 7 - 960 x - 4345 x - 797 x + 104 x + 1327 x + 862 x - 388 x 6 5 4 3 2 / 80 - 842 x + 86 x + 706 x - 544 x + 182 x - 30 x + 2) / (26431 x / 79 78 77 76 75 - 538428 x + 2710174 x + 3976483 x - 9685213 x - 291059848 x 74 73 72 71 - 133333381 x + 5863820626 x + 2002219187 x - 44223633294 x 70 69 68 67 - 37819213172 x + 160971312997 x + 363565384940 x - 39017108108 x 66 65 64 - 1022455917897 x - 1877453712485 x + 1351960721509 x 63 62 61 + 4859718938889 x + 2658856008540 x - 5579355349644 x 60 59 58 - 10010560577129 x - 748836015346 x + 12901716520710 x 57 56 55 + 8470785356098 x - 4104505355914 x - 9453166204848 x 54 53 52 - 5780673538679 x + 2082507641918 x + 6515262512581 x 51 50 49 + 4176538159686 x - 1367332692020 x - 4228331921150 x 48 47 46 - 1007326510711 x + 504754766591 x + 1201522309380 x 45 44 43 + 684047805435 x - 200048689627 x - 483298006623 x 42 41 40 39 - 105518133214 x + 158087412931 x - 32201526449 x + 24022549434 x 38 37 36 35 + 17406060362 x - 609563645 x - 7650820426 x - 5123788458 x 34 33 32 31 - 15279915 x + 1976963883 x + 728890128 x - 1070748155 x 30 29 28 27 + 923402947 x - 195797320 x - 191026106 x - 74502888 x 26 25 24 23 22 + 68699611 x + 71736972 x - 22827326 x - 40953825 x + 30609270 x 21 20 19 18 17 16 - 8278520 x + 805942 x - 2368 x + 14765 x + 21651 x + 13584 x 15 14 13 12 11 10 9 - 327 x - 8694 x - 10761 x + 3007 x + 2439 x + 2977 x - 212 x 8 7 6 5 4 3 2 - 1899 x - 720 x + 1080 x + 534 x - 999 x + 516 x - 134 x + 18 x - 1) and in Maple notation -x^2*(917*x^77-11376*x^76+32295*x^75-214814*x^74-127021*x^73+24159368*x^72-\ 47647205*x^71-356797187*x^70+329761349*x^69+2507526014*x^68+838820680*x^67-\ 12473599399*x^66-24463724767*x^65-19665809142*x^64+96843153613*x^63+ 165174024880*x^62-27632854558*x^61-390157991950*x^60-430075280392*x^59+ 226409252434*x^58+1011925317720*x^57+481444474683*x^56-672419513060*x^55-\ 1033886918259*x^54-431399887393*x^53+470927426947*x^52+876216115226*x^51+ 501321702377*x^50-240935777781*x^49-688447982122*x^48-240990587840*x^47+ 108174861719*x^46+241144623411*x^45+155481500342*x^44-12736907544*x^43-\ 101054185746*x^42-53845491245*x^41+24984998494*x^40+4886316375*x^39+6474065457* x^38+4697325822*x^37+671266433*x^36-1709454186*x^35-1761379266*x^34-560102825*x ^33+399498037*x^32+463334997*x^31-257947382*x^30+181888069*x^29+10415147*x^28-\ 45444863*x^27-42942779*x^26-778082*x^25+26752604*x^24+7645341*x^23-18420145*x^ 22+7370539*x^21-1021359*x^20-7059*x^19-1816*x^18+2994*x^17+5573*x^16+5293*x^15+ 2592*x^14-960*x^13-4345*x^12-797*x^11+104*x^10+1327*x^9+862*x^8-388*x^7-842*x^6 +86*x^5+706*x^4-544*x^3+182*x^2-30*x+2)/(26431*x^80-538428*x^79+2710174*x^78+ 3976483*x^77-9685213*x^76-291059848*x^75-133333381*x^74+5863820626*x^73+ 2002219187*x^72-44223633294*x^71-37819213172*x^70+160971312997*x^69+ 363565384940*x^68-39017108108*x^67-1022455917897*x^66-1877453712485*x^65+ 1351960721509*x^64+4859718938889*x^63+2658856008540*x^62-5579355349644*x^61-\ 10010560577129*x^60-748836015346*x^59+12901716520710*x^58+8470785356098*x^57-\ 4104505355914*x^56-9453166204848*x^55-5780673538679*x^54+2082507641918*x^53+ 6515262512581*x^52+4176538159686*x^51-1367332692020*x^50-4228331921150*x^49-\ 1007326510711*x^48+504754766591*x^47+1201522309380*x^46+684047805435*x^45-\ 200048689627*x^44-483298006623*x^43-105518133214*x^42+158087412931*x^41-\ 32201526449*x^40+24022549434*x^39+17406060362*x^38-609563645*x^37-7650820426*x^ 36-5123788458*x^35-15279915*x^34+1976963883*x^33+728890128*x^32-1070748155*x^31 +923402947*x^30-195797320*x^29-191026106*x^28-74502888*x^27+68699611*x^26+ 71736972*x^25-22827326*x^24-40953825*x^23+30609270*x^22-8278520*x^21+805942*x^ 20-2368*x^19+14765*x^18+21651*x^17+13584*x^16-327*x^15-8694*x^14-10761*x^13+ 3007*x^12+2439*x^11+2977*x^10-212*x^9-1899*x^8-720*x^7+1080*x^6+534*x^5-999*x^4 +516*x^3-134*x^2+18*x-1) Theorem number, 43, Let a(n) be the, 9, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = ---------------------------------------------- / 9 8 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x - x - x + 1 n = 0 and let b(n) be the, 9, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = ---------------------------------------------- / 9 8 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n 2 36 35 34 33 ) c(n) x = - 2 x (746384 x + 693376 x - 464120 x - 1595440 x / ----- n = 0 32 31 30 29 28 - 1395819 x + 342648 x + 2530266 x + 3287040 x - 752850 x 27 26 25 24 23 - 1552436 x - 1120123 x - 411856 x + 134171 x + 377708 x 22 21 20 19 18 17 + 333954 x + 84528 x - 195687 x + 7752 x - 2640 x - 4132 x 16 15 14 13 12 11 10 - 2847 x - 806 x + 1004 x + 1663 x + 486 x - 1662 x + 501 x 9 8 7 6 5 4 3 2 / + 16 x + 17 x + 9 x - 2 x - 11 x - 12 x - x + 14 x - 7 x + 1) / / 9 8 7 6 5 4 3 2 ((512 x + 256 x + 128 x + 64 x + 32 x + 16 x + 8 x + 4 x + 2 x - 1) 36 35 34 33 32 31 30 (16 x - 96 x - 40 x + 680 x + 1545 x - 1976 x - 18270 x 29 28 27 26 25 24 - 32592 x + 70134 x + 51764 x - 20055 x - 76680 x - 64533 x 23 22 21 20 19 18 + 13512 x + 86490 x + 59680 x - 69339 x - 8640 x - 6060 x 17 16 15 14 13 12 + 328 x + 4669 x + 4684 x + 1249 x - 2288 x - 2090 x 11 10 9 8 7 6 4 3 + 2028 x - 537 x - 16 x + 18 x + 36 x + 27 x - 21 x - 12 x 2 + 21 x - 8 x + 1)) and in Maple notation -2*x^2*(746384*x^36+693376*x^35-464120*x^34-1595440*x^33-1395819*x^32+342648*x^ 31+2530266*x^30+3287040*x^29-752850*x^28-1552436*x^27-1120123*x^26-411856*x^25+ 134171*x^24+377708*x^23+333954*x^22+84528*x^21-195687*x^20+7752*x^19-2640*x^18-\ 4132*x^17-2847*x^16-806*x^15+1004*x^14+1663*x^13+486*x^12-1662*x^11+501*x^10+16 *x^9+17*x^8+9*x^7-2*x^6-11*x^5-12*x^4-x^3+14*x^2-7*x+1)/(512*x^9+256*x^8+128*x^ 7+64*x^6+32*x^5+16*x^4+8*x^3+4*x^2+2*x-1)/(16*x^36-96*x^35-40*x^34+680*x^33+ 1545*x^32-1976*x^31-18270*x^30-32592*x^29+70134*x^28+51764*x^27-20055*x^26-\ 76680*x^25-64533*x^24+13512*x^23+86490*x^22+59680*x^21-69339*x^20-8640*x^19-\ 6060*x^18+328*x^17+4669*x^16+4684*x^15+1249*x^14-2288*x^13-2090*x^12+2028*x^11-\ 537*x^10-16*x^9+18*x^8+36*x^7+27*x^6-21*x^4-12*x^3+21*x^2-8*x+1) Theorem number, 44, Let a(n) be the, 9, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = ---------------------------------------------- / 9 8 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x - x - x + 1 n = 0 and let b(n) be the, 10, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = ---------------------------------------------------- / 10 9 8 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n 2 87 86 85 84 ) c(n) x = x (2102 x + 52719 x + 597117 x + 924233 x / ----- n = 0 83 82 81 80 - 14148101 x - 32439212 x + 95198553 x + 508847554 x 79 78 77 76 + 380202246 x - 4480543591 x - 28283048296 x + 70078432867 x 75 74 73 + 150655511008 x - 167195525148 x - 732367718110 x 72 71 70 - 335599771526 x + 2082775307816 x + 4415073580034 x 69 68 67 + 150526434480 x - 11000966570692 x - 11705916918888 x 66 65 64 + 4767746571403 x + 23173641307722 x + 20722359187031 x 63 62 61 - 7732193707436 x - 37369048913640 x - 30105971646182 x 60 59 58 + 19223316602225 x + 37290499204948 x + 19656110178658 x 57 56 55 - 10193282016902 x - 25121667404586 x - 16043394306438 x 54 53 52 + 3914589252894 x + 14305399808086 x + 8160285506280 x 51 50 49 - 1502050813993 x - 3671562017201 x - 3372655473711 x 48 47 46 - 1204072355707 x + 907948733567 x + 1526076637877 x 45 44 43 + 669821117167 x - 399155473790 x - 481712058067 x 42 41 40 39 + 109286185614 x - 38419017359 x + 13291440771 x + 25127317681 x 38 37 36 35 + 14303459195 x - 302303781 x - 5983648326 x - 2425656299 x 34 33 32 31 + 1611549496 x + 169354871 x - 1343347711 x + 429179051 x 30 29 28 27 + 333692262 x + 183422082 x - 52044564 x - 150297453 x 26 25 24 23 22 - 56264623 x + 71789387 x + 46828994 x - 57730606 x + 19490939 x 21 20 19 18 17 16 - 2333525 x - 18832 x - 26352 x - 20748 x - 7006 x + 6663 x 15 14 13 12 11 10 + 11083 x + 5105 x - 483 x - 1892 x - 3338 x - 1094 x 9 8 7 6 5 4 3 2 + 1284 x + 1690 x + 52 x - 1244 x - 230 x + 1096 x - 698 x + 210 x / 90 89 88 87 - 32 x + 2) / (127177 x + 1097109 x + 5547735 x + 1402754 x / 86 85 84 83 - 189393471 x - 550658639 x + 2351583902 x + 9662293418 x 82 81 80 79 - 4538670363 x - 93524697698 x - 179896951505 x + 350804669540 x 78 77 76 + 2982220869104 x - 2386099742098 x - 11525616726817 x 75 74 73 - 3439880248093 x + 30174541830924 x + 47708838927361 x 72 71 70 - 19003993799677 x - 137369443912331 x - 100026367721369 x 69 68 67 + 151677576630909 x + 271823244208334 x + 75108975489053 x 66 65 64 - 267067159316947 x - 372696835178863 x - 51932888546216 x 63 62 61 + 387508681970393 x + 378194590147246 x - 116505765649432 x 60 59 58 - 285621225189174 x - 186865050696511 x + 36652976641451 x 57 56 55 + 166317062953803 x + 112579520703202 x - 22311051941223 x 54 53 52 - 80990377093417 x - 35778535250355 x + 9258137992585 x 51 50 49 + 13204340753404 x + 15650913525176 x + 5036286810213 x 48 47 46 - 5238084122396 x - 6667475463498 x - 1308063102089 x 45 44 43 + 2631914672301 x + 1177826106067 x - 991868805984 x 42 41 40 39 + 420768138293 x - 82973041816 x - 90016072044 x - 31789873032 x 38 37 36 35 + 18478470668 x + 20894451699 x - 1624376723 x - 8086743347 x 34 33 32 31 + 3641157224 x + 4178435720 x - 3036442693 x - 824956075 x 30 29 28 27 - 181813182 x + 486357273 x + 393100789 x - 83941201 x 26 25 24 23 - 253112322 x + 10681039 x + 144101630 x - 85545891 x 22 21 20 19 18 + 20271374 x - 1800802 x + 84362 x + 63515 x + 3775 x 17 16 15 14 13 12 - 39850 x - 32842 x - 1126 x + 9146 x + 7385 x + 5789 x 11 10 9 8 7 6 5 - 2592 x - 4623 x - 1403 x + 2193 x + 1734 x - 1146 x - 1134 x 4 3 2 + 1398 x - 633 x + 151 x - 19 x + 1) and in Maple notation x^2*(2102*x^87+52719*x^86+597117*x^85+924233*x^84-14148101*x^83-32439212*x^82+ 95198553*x^81+508847554*x^80+380202246*x^79-4480543591*x^78-28283048296*x^77+ 70078432867*x^76+150655511008*x^75-167195525148*x^74-732367718110*x^73-\ 335599771526*x^72+2082775307816*x^71+4415073580034*x^70+150526434480*x^69-\ 11000966570692*x^68-11705916918888*x^67+4767746571403*x^66+23173641307722*x^65+ 20722359187031*x^64-7732193707436*x^63-37369048913640*x^62-30105971646182*x^61+ 19223316602225*x^60+37290499204948*x^59+19656110178658*x^58-10193282016902*x^57 -25121667404586*x^56-16043394306438*x^55+3914589252894*x^54+14305399808086*x^53 +8160285506280*x^52-1502050813993*x^51-3671562017201*x^50-3372655473711*x^49-\ 1204072355707*x^48+907948733567*x^47+1526076637877*x^46+669821117167*x^45-\ 399155473790*x^44-481712058067*x^43+109286185614*x^42-38419017359*x^41+ 13291440771*x^40+25127317681*x^39+14303459195*x^38-302303781*x^37-5983648326*x^ 36-2425656299*x^35+1611549496*x^34+169354871*x^33-1343347711*x^32+429179051*x^ 31+333692262*x^30+183422082*x^29-52044564*x^28-150297453*x^27-56264623*x^26+ 71789387*x^25+46828994*x^24-57730606*x^23+19490939*x^22-2333525*x^21-18832*x^20 -26352*x^19-20748*x^18-7006*x^17+6663*x^16+11083*x^15+5105*x^14-483*x^13-1892*x ^12-3338*x^11-1094*x^10+1284*x^9+1690*x^8+52*x^7-1244*x^6-230*x^5+1096*x^4-698* x^3+210*x^2-32*x+2)/(127177*x^90+1097109*x^89+5547735*x^88+1402754*x^87-\ 189393471*x^86-550658639*x^85+2351583902*x^84+9662293418*x^83-4538670363*x^82-\ 93524697698*x^81-179896951505*x^80+350804669540*x^79+2982220869104*x^78-\ 2386099742098*x^77-11525616726817*x^76-3439880248093*x^75+30174541830924*x^74+ 47708838927361*x^73-19003993799677*x^72-137369443912331*x^71-100026367721369*x^ 70+151677576630909*x^69+271823244208334*x^68+75108975489053*x^67-\ 267067159316947*x^66-372696835178863*x^65-51932888546216*x^64+387508681970393*x ^63+378194590147246*x^62-116505765649432*x^61-285621225189174*x^60-\ 186865050696511*x^59+36652976641451*x^58+166317062953803*x^57+112579520703202*x ^56-22311051941223*x^55-80990377093417*x^54-35778535250355*x^53+9258137992585*x ^52+13204340753404*x^51+15650913525176*x^50+5036286810213*x^49-5238084122396*x^ 48-6667475463498*x^47-1308063102089*x^46+2631914672301*x^45+1177826106067*x^44-\ 991868805984*x^43+420768138293*x^42-82973041816*x^41-90016072044*x^40-\ 31789873032*x^39+18478470668*x^38+20894451699*x^37-1624376723*x^36-8086743347*x ^35+3641157224*x^34+4178435720*x^33-3036442693*x^32-824956075*x^31-181813182*x^ 30+486357273*x^29+393100789*x^28-83941201*x^27-253112322*x^26+10681039*x^25+ 144101630*x^24-85545891*x^23+20271374*x^22-1800802*x^21+84362*x^20+63515*x^19+ 3775*x^18-39850*x^17-32842*x^16-1126*x^15+9146*x^14+7385*x^13+5789*x^12-2592*x^ 11-4623*x^10-1403*x^9+2193*x^8+1734*x^7-1146*x^6-1134*x^5+1398*x^4-633*x^3+151* x^2-19*x+1) Theorem number, 45, Let a(n) be the, 10, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = ---------------------------------------------------- / 10 9 8 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x - x - x - x + 1 n = 0 and let b(n) be the, 10, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) b(n) x = ---------------------------------------------------- / 10 9 8 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x - x - x - x + 1 n = 0 Let c(n) be the binomial convolution n ----- \ c(n) = ) binomial(n, r) a(r) b(n - r) / ----- r = 0 Then the generating function of the sequence c(n) is infinity ----- \ n 2 49 48 47 46 ) c(n) x = 2 x (256000 x + 198400 x - 910048 x - 2052708 x / ----- n = 0 45 44 43 42 41 + 32190 x + 6691526 x + 9910708 x - 9323270 x - 86297046 x 40 39 38 37 36 - 39905900 x + 53427758 x + 96740236 x + 59940494 x - 19254286 x 35 34 33 32 31 - 79288112 x - 78978815 x - 16785013 x + 63456146 x + 25463705 x 30 29 28 27 26 + 6154849 x - 3248688 x - 6138537 x - 5071546 x - 2304527 x 25 24 23 22 21 + 410738 x + 1841097 x + 1234569 x - 931672 x + 58972 x 20 19 18 17 16 15 + 15042 x - 2186 x - 9102 x - 10148 x - 6657 x - 485 x 14 13 12 11 10 9 8 7 + 4463 x + 3214 x - 4409 x + 911 x - 32 x + 8 x + 26 x + 27 x 6 5 4 3 2 / 10 9 + 14 x - 6 x - 19 x - 9 x + 20 x - 8 x + 1) / ((1024 x + 512 x / 8 7 6 5 4 3 2 45 + 256 x + 128 x + 64 x + 32 x + 16 x + 8 x + 4 x + 2 x - 1) (x 44 43 42 41 40 39 + 15 x + 160 x + 1595 x + 15785 x - 43712 x - 52157 x 38 37 36 35 34 + 77974 x + 240497 x + 100061 x - 562181 x - 1141611 x 33 32 31 30 29 - 106552 x + 2814416 x + 392004 x - 1045984 x - 1355487 x 28 27 26 25 24 - 633678 x + 438405 x + 990838 x + 580887 x - 367582 x 23 22 21 20 19 18 - 702398 x + 273153 x - 10960 x + 26295 x + 30093 x + 15304 x 17 16 15 14 13 12 - 3177 x - 12708 x - 8325 x + 3408 x + 7191 x - 5307 x 11 10 9 8 7 6 5 4 + 1002 x + 80 x + 78 x + 26 x - 28 x - 46 x - 19 x + 23 x 3 2 + 26 x - 28 x + 9 x - 1)) and in Maple notation 2*x^2*(256000*x^49+198400*x^48-910048*x^47-2052708*x^46+32190*x^45+6691526*x^44 +9910708*x^43-9323270*x^42-86297046*x^41-39905900*x^40+53427758*x^39+96740236*x ^38+59940494*x^37-19254286*x^36-79288112*x^35-78978815*x^34-16785013*x^33+ 63456146*x^32+25463705*x^31+6154849*x^30-3248688*x^29-6138537*x^28-5071546*x^27 -2304527*x^26+410738*x^25+1841097*x^24+1234569*x^23-931672*x^22+58972*x^21+ 15042*x^20-2186*x^19-9102*x^18-10148*x^17-6657*x^16-485*x^15+4463*x^14+3214*x^ 13-4409*x^12+911*x^11-32*x^10+8*x^9+26*x^8+27*x^7+14*x^6-6*x^5-19*x^4-9*x^3+20* x^2-8*x+1)/(1024*x^10+512*x^9+256*x^8+128*x^7+64*x^6+32*x^5+16*x^4+8*x^3+4*x^2+ 2*x-1)/(x^45+15*x^44+160*x^43+1595*x^42+15785*x^41-43712*x^40-52157*x^39+77974* x^38+240497*x^37+100061*x^36-562181*x^35-1141611*x^34-106552*x^33+2814416*x^32+ 392004*x^31-1045984*x^30-1355487*x^29-633678*x^28+438405*x^27+990838*x^26+ 580887*x^25-367582*x^24-702398*x^23+273153*x^22-10960*x^21+26295*x^20+30093*x^ 19+15304*x^18-3177*x^17-12708*x^16-8325*x^15+3408*x^14+7191*x^13-5307*x^12+1002 *x^11+80*x^10+78*x^9+26*x^8-28*x^7-46*x^6-19*x^5+23*x^4+26*x^3-28*x^2+9*x-1) ------------------------- This ends this article that took, 43.705, seconds to produce.