Convolution Identities for the k-Bonacci numbers from k=2 (the Fibonacci case) all the way to k=, 20 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 Let b(n) be the binomial convolution n ----- \ b(n) = ) binomial(n, r) a(r) a(n - r) / ----- r = 0 Then the generating function of the sequence b(n) is infinity ----- 2 \ n 2 x ) b(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, 3, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = ---------------- / 3 2 ----- -x - x - x + 1 n = 0 Let b(n) be the binomial convolution n ----- \ b(n) = ) binomial(n, r) a(r) a(n - r) / ----- r = 0 Then the generating function of the sequence b(n) is infinity ----- 2 2 \ n 2 x (2 x - 1) (x + x + 1) ) b(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, 3, 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 Let b(n) be the binomial convolution n ----- \ b(n) = ) binomial(n, r) a(r) a(n - r) / ----- r = 0 Then the generating function of the sequence b(n) is infinity ----- \ n ) b(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, 4, 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 Let b(n) be the binomial convolution n ----- \ b(n) = ) binomial(n, r) a(r) a(n - r) / ----- r = 0 Then the generating function of the sequence b(n) is infinity ----- \ n 2 10 9 8 6 4 3 ) b(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, 5, 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 Let b(n) be the binomial convolution n ----- \ b(n) = ) binomial(n, r) a(r) a(n - r) / ----- r = 0 Then the generating function of the sequence b(n) is infinity ----- \ n 2 17 16 15 14 13 12 ) b(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, 6, 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 Let b(n) be the binomial convolution n ----- \ b(n) = ) binomial(n, r) a(r) a(n - r) / ----- r = 0 Then the generating function of the sequence b(n) is infinity ----- \ n 2 21 20 19 18 17 ) b(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, 7, 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 Let b(n) be the binomial convolution n ----- \ b(n) = ) binomial(n, r) a(r) a(n - r) / ----- r = 0 Then the generating function of the sequence b(n) is infinity ----- \ n 2 31 30 29 28 27 ) b(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, 8, 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 Let b(n) be the binomial convolution n ----- \ b(n) = ) binomial(n, r) a(r) a(n - r) / ----- r = 0 Then the generating function of the sequence b(n) is infinity ----- \ n 2 36 35 34 33 ) b(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, 9, 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 Let b(n) be the binomial convolution n ----- \ b(n) = ) binomial(n, r) a(r) a(n - r) / ----- r = 0 Then the generating function of the sequence b(n) is infinity ----- \ n 2 49 48 47 46 ) b(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) Theorem number, 10, Let a(n) be the, 11, -bonacci numbers that are defined via the generating function infinity ----- \ n x ) a(n) x = ---------------------------------------------------------- / 11 10 9 8 7 6 5 4 3 2 ----- -x - x - x - x - x - x - x - x - x - x - x + 1 n = 0 Let b(n) be the binomial convolution n ----- \ b(n) = ) binomial(n, r) a(r) a(n - r) / ----- r = 0 Then the generating function of the sequence b(n) is infinity ----- \ n 2 55 54 53 ) b(n) x = 2 x (242944928 x + 186456720 x - 416025200 x / ----- n = 0 52 51 50 49 - 911624624 x - 348304424 x + 1237294536 x + 2251248296 x 48 47 46 45 + 727001572 x - 3481433868 x - 6985324764 x + 361950182 x 44 43 42 41 + 4960169370 x + 4531050138 x + 1286236729 x - 1779888451 x 40 39 38 37 - 2888486849 x - 1959821606 x - 75885709 x + 1364500907 x 36 35 34 33 + 1309173503 x - 250622768 x - 170365206 x - 135054534 x 32 31 30 29 28 - 87611338 x - 36701906 x + 2004529 x + 20913049 x + 20862701 x 27 26 25 24 23 + 8362062 x - 6014543 x - 9247205 x + 4613257 x - 131296 x 22 21 20 19 18 17 - 63400 x - 40586 x - 20422 x + 1930 x + 19381 x + 23191 x 16 15 14 13 12 11 + 10481 x - 9086 x - 13393 x + 10593 x - 1859 x + 132 x 10 9 8 7 6 5 4 3 2 + 86 x + 38 x - 6 x - 34 x - 35 x - 9 x + 23 x + 22 x - 27 x / 11 10 9 8 7 6 + 9 x - 1) / ((2048 x + 1024 x + 512 x + 256 x + 128 x + 64 x / 5 4 3 2 55 54 53 + 32 x + 16 x + 8 x + 4 x + 2 x - 1) (32 x - 320 x + 480 x 52 51 50 49 48 47 + 3120 x - 320 x - 33968 x - 89656 x + 141328 x + 1561912 x 46 45 44 43 42 + 3202540 x - 9599744 x - 7606284 x + 8237538 x + 20548764 x 41 40 39 38 37 + 10371990 x - 21052397 x - 42714000 x - 17678164 x + 46405720 x 36 35 34 33 32 + 67570811 x - 33214060 x - 28792236 x - 16838516 x - 189358 x 31 30 29 28 27 + 11366484 x + 12110192 x + 4044588 x - 4965815 x - 6965262 x 26 25 24 23 22 - 819648 x + 4575086 x - 1418241 x + 243056 x + 107352 x 21 20 19 18 17 16 - 3200 x - 64012 x - 63406 x - 20556 x + 23014 x + 30045 x 15 14 13 12 11 10 9 - 948 x - 23208 x + 12812 x - 1683 x + 132 x - 40 x - 116 x 8 7 6 5 4 3 2 - 90 x - 8 x + 56 x + 48 x - 17 x - 46 x + 36 x - 10 x + 1)) and in Maple notation 2*x^2*(242944928*x^55+186456720*x^54-416025200*x^53-911624624*x^52-348304424*x^ 51+1237294536*x^50+2251248296*x^49+727001572*x^48-3481433868*x^47-6985324764*x^ 46+361950182*x^45+4960169370*x^44+4531050138*x^43+1286236729*x^42-1779888451*x^ 41-2888486849*x^40-1959821606*x^39-75885709*x^38+1364500907*x^37+1309173503*x^ 36-250622768*x^35-170365206*x^34-135054534*x^33-87611338*x^32-36701906*x^31+ 2004529*x^30+20913049*x^29+20862701*x^28+8362062*x^27-6014543*x^26-9247205*x^25 +4613257*x^24-131296*x^23-63400*x^22-40586*x^21-20422*x^20+1930*x^19+19381*x^18 +23191*x^17+10481*x^16-9086*x^15-13393*x^14+10593*x^13-1859*x^12+132*x^11+86*x^ 10+38*x^9-6*x^8-34*x^7-35*x^6-9*x^5+23*x^4+22*x^3-27*x^2+9*x-1)/(2048*x^11+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)/(32*x^55-\ 320*x^54+480*x^53+3120*x^52-320*x^51-33968*x^50-89656*x^49+141328*x^48+1561912* x^47+3202540*x^46-9599744*x^45-7606284*x^44+8237538*x^43+20548764*x^42+10371990 *x^41-21052397*x^40-42714000*x^39-17678164*x^38+46405720*x^37+67570811*x^36-\ 33214060*x^35-28792236*x^34-16838516*x^33-189358*x^32+11366484*x^31+12110192*x^ 30+4044588*x^29-4965815*x^28-6965262*x^27-819648*x^26+4575086*x^25-1418241*x^24 +243056*x^23+107352*x^22-3200*x^21-64012*x^20-63406*x^19-20556*x^18+23014*x^17+ 30045*x^16-948*x^15-23208*x^14+12812*x^13-1683*x^12+132*x^11-40*x^10-116*x^9-90 *x^8-8*x^7+56*x^6+48*x^5-17*x^4-46*x^3+36*x^2-10*x+1) Theorem number, 11, Let a(n) be the, 12, -bonacci numbers that are defined via the generating function infinity ----- \ n ) a(n) x = / ----- n = 0 x ---------------------------------------------------------------- 12 11 10 9 8 7 6 5 4 3 2 -x - x - x - x - x - x - x - x - x - x - x - x + 1 Let b(n) be the binomial convolution n ----- \ b(n) = ) binomial(n, r) a(r) a(n - r) / ----- r = 0 Then the generating function of the sequence b(n) is infinity ----- \ n 2 71 70 69 ) b(n) x = 2 x (21233664 x - 1327104 x - 136088576 x / ----- n = 0 68 67 66 65 - 169234240 x + 399408948 x + 1384292382 x + 482892508 x 64 63 62 61 - 5114256908 x - 10105616794 x + 9377948398 x + 112341766524 x 60 59 58 57 + 49133228678 x - 134543773782 x - 216562037168 x - 71154372838 x 56 55 54 + 190259923750 x + 315745322626 x + 148755666482 x 53 52 51 - 209537180405 x - 449293513624 x - 272647877342 x 50 49 48 47 + 302190195499 x + 297161210099 x + 118603272284 x - 41042878581 x 46 45 44 43 - 109144003721 x - 91911751444 x - 33550095497 x + 19754164041 x 42 41 40 39 + 41155237281 x + 28409864990 x - 792075616 x - 17457775903 x 38 37 36 35 - 303672688 x - 738199696 x - 237200318 x + 216195391 x 34 33 32 31 + 394649596 x + 340186707 x + 170374634 x - 1664824 x 30 29 28 27 26 - 97043064 x - 84482925 x + 3187486 x + 59719091 x - 21519262 x 25 24 23 22 21 20 - 131616 x - 176064 x - 53684 x + 52214 x + 88734 x + 57907 x 19 18 17 16 15 14 - 7235 x - 57151 x - 50614 x + 7928 x + 44130 x - 24644 x 13 12 11 10 9 8 7 6 + 4171 x + 12 x - 66 x - 106 x - 96 x - 46 x + 16 x + 52 x 5 4 3 2 / 12 11 + 35 x - 20 x - 41 x + 35 x - 10 x + 1) / ((4096 x + 2048 x / 10 9 8 7 6 5 4 3 + 1024 x + 512 x + 256 x + 128 x + 64 x + 32 x + 16 x + 8 x 2 66 65 64 63 62 + 4 x + 2 x - 1) (x + 18 x + 228 x + 2691 x + 31525 x 61 60 59 58 57 + 369512 x - 1639601 x - 960661 x + 5814818 x + 9920087 x 56 55 54 53 - 9671007 x - 54049626 x - 44940667 x + 150035531 x 52 51 50 49 + 431930216 x + 106149072 x - 1414123968 x - 396181968 x 48 47 46 45 + 970386099 x + 1362908514 x + 407632383 x - 1068120238 x 44 43 42 41 - 1657574276 x - 636186266 x + 1163738857 x + 1825645214 x 40 39 38 37 + 284171949 x - 1607854384 x - 174713344 x - 13407612 x 36 35 34 33 + 160416532 x + 198111898 x + 112560836 x - 10268563 x 32 31 30 29 28 - 83002373 x - 69476723 x - 2739662 x + 43023321 x + 19976449 x 27 26 25 24 23 - 27251655 x + 8639004 x - 166048 x - 393482 x - 397854 x 22 21 20 19 18 17 - 200658 x + 36930 x + 159528 x + 115332 x - 16635 x - 89326 x 16 15 14 13 12 11 - 23660 x + 68826 x - 28685 x + 3784 x - 408 x - 318 x 10 9 8 7 6 5 4 3 2 - 106 x + 86 x + 150 x + 76 x - 44 x - 84 x - 2 x + 73 x - 45 x + 11 x - 1)) and in Maple notation 2*x^2*(21233664*x^71-1327104*x^70-136088576*x^69-169234240*x^68+399408948*x^67+ 1384292382*x^66+482892508*x^65-5114256908*x^64-10105616794*x^63+9377948398*x^62 +112341766524*x^61+49133228678*x^60-134543773782*x^59-216562037168*x^58-\ 71154372838*x^57+190259923750*x^56+315745322626*x^55+148755666482*x^54-\ 209537180405*x^53-449293513624*x^52-272647877342*x^51+302190195499*x^50+ 297161210099*x^49+118603272284*x^48-41042878581*x^47-109144003721*x^46-\ 91911751444*x^45-33550095497*x^44+19754164041*x^43+41155237281*x^42+28409864990 *x^41-792075616*x^40-17457775903*x^39-303672688*x^38-738199696*x^37-237200318*x ^36+216195391*x^35+394649596*x^34+340186707*x^33+170374634*x^32-1664824*x^31-\ 97043064*x^30-84482925*x^29+3187486*x^28+59719091*x^27-21519262*x^26-131616*x^ 25-176064*x^24-53684*x^23+52214*x^22+88734*x^21+57907*x^20-7235*x^19-57151*x^18 -50614*x^17+7928*x^16+44130*x^15-24644*x^14+4171*x^13+12*x^12-66*x^11-106*x^10-\ 96*x^9-46*x^8+16*x^7+52*x^6+35*x^5-20*x^4-41*x^3+35*x^2-10*x+1)/(4096*x^12+2048 *x^11+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^66+18*x^65+228*x^64+2691*x^63+31525*x^62+369512*x^61-1639601*x^60-960661*x^ 59+5814818*x^58+9920087*x^57-9671007*x^56-54049626*x^55-44940667*x^54+150035531 *x^53+431930216*x^52+106149072*x^51-1414123968*x^50-396181968*x^49+970386099*x^ 48+1362908514*x^47+407632383*x^46-1068120238*x^45-1657574276*x^44-636186266*x^ 43+1163738857*x^42+1825645214*x^41+284171949*x^40-1607854384*x^39-174713344*x^ 38-13407612*x^37+160416532*x^36+198111898*x^35+112560836*x^34-10268563*x^33-\ 83002373*x^32-69476723*x^31-2739662*x^30+43023321*x^29+19976449*x^28-27251655*x ^27+8639004*x^26-166048*x^25-393482*x^24-397854*x^23-200658*x^22+36930*x^21+ 159528*x^20+115332*x^19-16635*x^18-89326*x^17-23660*x^16+68826*x^15-28685*x^14+ 3784*x^13-408*x^12-318*x^11-106*x^10+86*x^9+150*x^8+76*x^7-44*x^6-84*x^5-2*x^4+ 73*x^3-45*x^2+11*x-1) Theorem number, 12, Let a(n) be the, 13, -bonacci numbers that are defined via the generating function infinity ----- \ n ) a(n) x = / ----- n = 0 x ---------------------------------------------------------------------- 13 12 11 10 9 8 7 6 5 4 3 2 -x - x - x - x - x - x - x - x - x - x - x - x - x + 1 Let b(n) be the binomial convolution n ----- \ b(n) = ) binomial(n, r) a(r) a(n - r) / ----- r = 0 Then the generating function of the sequence b(n) is infinity ----- \ n 2 78 77 ) b(n) x = - 2 x (126168838464 x + 53219551872 x / ----- n = 0 76 75 74 - 401747573424 x - 627578478464 x + 281079072908 x 73 72 71 + 1931686295320 x + 1870453617871 x - 2059920416216 x 70 69 68 - 7145517421107 x - 4727284063944 x + 12094824746004 x 67 66 65 + 33029131010616 x - 69402833474 x - 35409290094424 x 64 63 62 - 34868904519350 x - 2707177356488 x + 30053015504256 x 61 60 59 + 36251470175976 x + 13249383166161 x - 18451361971493 x 58 57 56 - 33283085257686 x - 20129921447779 x + 9281005486493 x 55 54 53 + 26224244832916 x + 6690166160626 x - 2308104224530 x 52 51 50 - 5101376053706 x - 4173750451998 x - 1812572753028 x 49 48 47 + 264140938548 x + 1244637923408 x + 1145083736630 x 46 45 44 + 464197021716 x - 199286834416 x - 431587992592 x 43 42 41 40 - 181377649622 x + 157436277856 x - 14771300128 x + 1892190638 x 39 38 37 36 + 5796741050 x + 4649242306 x + 2047361122 x - 198188676 x 35 34 33 32 - 1341684228 x - 1345126596 x - 614896866 x + 217968882 x 31 30 29 28 + 547111434 x + 162473690 x - 350444952 x + 93289508 x 27 26 25 24 23 + 237068 x + 1125822 x + 1054338 x + 631322 x + 163198 x 22 21 20 19 18 - 164004 x - 254844 x - 126552 x + 80842 x + 166536 x 17 16 15 14 13 12 + 32640 x - 128615 x + 57902 x - 8333 x - 312 x - 258 x 11 10 9 8 7 6 5 4 - 114 x + 32 x + 118 x + 114 x + 36 x - 52 x - 70 x + 5 x 3 2 / 13 12 11 + 67 x - 44 x + 11 x - 1) / ((8192 x + 4096 x + 2048 x / 10 9 8 7 6 5 4 3 + 1024 x + 512 x + 256 x + 128 x + 64 x + 32 x + 16 x + 8 x 2 78 77 76 75 74 + 4 x + 2 x - 1) (64 x - 960 x + 3696 x + 6944 x - 38164 x 73 72 71 70 69 - 162876 x + 81809 x + 2783996 x + 8336731 x - 17508296 x 68 67 66 65 - 207160356 x - 447642624 x + 1983510178 x + 1252500576 x 64 63 62 61 - 3420559418 x - 6084208856 x + 556808548 x + 14064325592 x 60 59 58 57 + 16497395320 x - 9982794196 x - 49295845075 x - 40984116676 x 56 55 54 53 + 53988369643 x + 130636215424 x - 32649598072 x - 89094615376 x 52 51 50 49 - 63639102144 x + 5779455280 x + 60684918488 x + 60519500064 x 48 47 46 45 + 11315731984 x - 41467961936 x - 51049967720 x - 11069180680 x 44 43 42 41 + 33576539786 x + 28265008664 x - 15357998370 x - 1269538608 x 40 39 38 37 - 3014421904 x - 1995441040 x - 248531180 x + 955962256 x 36 35 34 33 + 1133401860 x + 524886768 x - 215404344 x - 499139104 x 32 31 30 29 - 216259392 x + 194889448 x + 183006810 x - 165787528 x 28 27 26 25 24 + 40958086 x + 2483520 x + 1508824 x + 69520 x - 821136 x 23 22 21 20 19 - 858352 x - 314536 x + 252768 x + 398192 x + 101872 x 18 17 16 15 14 13 - 222388 x - 137660 x + 186869 x - 64724 x + 8983 x + 312 x 12 11 10 9 8 7 6 5 - 204 x - 432 x - 326 x - 48 x + 166 x + 168 x + 4 x - 120 x 4 3 2 - 40 x + 108 x - 55 x + 12 x - 1)) and in Maple notation -2*x^2*(126168838464*x^78+53219551872*x^77-401747573424*x^76-627578478464*x^75+ 281079072908*x^74+1931686295320*x^73+1870453617871*x^72-2059920416216*x^71-\ 7145517421107*x^70-4727284063944*x^69+12094824746004*x^68+33029131010616*x^67-\ 69402833474*x^66-35409290094424*x^65-34868904519350*x^64-2707177356488*x^63+ 30053015504256*x^62+36251470175976*x^61+13249383166161*x^60-18451361971493*x^59 -33283085257686*x^58-20129921447779*x^57+9281005486493*x^56+26224244832916*x^55 +6690166160626*x^54-2308104224530*x^53-5101376053706*x^52-4173750451998*x^51-\ 1812572753028*x^50+264140938548*x^49+1244637923408*x^48+1145083736630*x^47+ 464197021716*x^46-199286834416*x^45-431587992592*x^44-181377649622*x^43+ 157436277856*x^42-14771300128*x^41+1892190638*x^40+5796741050*x^39+4649242306*x ^38+2047361122*x^37-198188676*x^36-1341684228*x^35-1345126596*x^34-614896866*x^ 33+217968882*x^32+547111434*x^31+162473690*x^30-350444952*x^29+93289508*x^28+ 237068*x^27+1125822*x^26+1054338*x^25+631322*x^24+163198*x^23-164004*x^22-\ 254844*x^21-126552*x^20+80842*x^19+166536*x^18+32640*x^17-128615*x^16+57902*x^ 15-8333*x^14-312*x^13-258*x^12-114*x^11+32*x^10+118*x^9+114*x^8+36*x^7-52*x^6-\ 70*x^5+5*x^4+67*x^3-44*x^2+11*x-1)/(8192*x^13+4096*x^12+2048*x^11+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)/(64*x^78-960*x^77+ 3696*x^76+6944*x^75-38164*x^74-162876*x^73+81809*x^72+2783996*x^71+8336731*x^70 -17508296*x^69-207160356*x^68-447642624*x^67+1983510178*x^66+1252500576*x^65-\ 3420559418*x^64-6084208856*x^63+556808548*x^62+14064325592*x^61+16497395320*x^ 60-9982794196*x^59-49295845075*x^58-40984116676*x^57+53988369643*x^56+ 130636215424*x^55-32649598072*x^54-89094615376*x^53-63639102144*x^52+5779455280 *x^51+60684918488*x^50+60519500064*x^49+11315731984*x^48-41467961936*x^47-\ 51049967720*x^46-11069180680*x^45+33576539786*x^44+28265008664*x^43-15357998370 *x^42-1269538608*x^41-3014421904*x^40-1995441040*x^39-248531180*x^38+955962256* x^37+1133401860*x^36+524886768*x^35-215404344*x^34-499139104*x^33-216259392*x^ 32+194889448*x^31+183006810*x^30-165787528*x^29+40958086*x^28+2483520*x^27+ 1508824*x^26+69520*x^25-821136*x^24-858352*x^23-314536*x^22+252768*x^21+398192* x^20+101872*x^19-222388*x^18-137660*x^17+186869*x^16-64724*x^15+8983*x^14+312*x ^13-204*x^12-432*x^11-326*x^10-48*x^9+166*x^8+168*x^7+4*x^6-120*x^5-40*x^4+108* x^3-55*x^2+12*x-1) Theorem number, 13, Let a(n) be the, 14, -bonacci numbers that are defined via the generating function infinity ----- \ n 14 13 12 11 10 9 8 7 6 5 ) a(n) x = x/(-x - x - x - x - x - x - x - x - x - x / ----- n = 0 4 3 2 - x - x - x - x + 1) Let b(n) be the binomial convolution n ----- \ b(n) = ) binomial(n, r) a(r) a(n - r) / ----- r = 0 Then the generating function of the sequence b(n) is infinity ----- \ n 2 97 96 95 ) b(n) x = 2 x (2202927104 x - 2714320896 x - 19665479680 x / ----- n = 0 94 93 92 91 - 2217411840 x + 119072541792 x + 194797171244 x - 366352500514 x 90 89 88 - 1721204784326 x - 923005206840 x + 7930254955184 x 87 86 85 + 18376496787552 x - 21096967369576 x - 268978879923960 x 84 83 82 - 76781535909108 x + 541451942216572 x + 730522973671272 x 81 80 79 - 124100301220312 x - 1374453998368544 x - 1454445181913474 x 78 77 76 + 409365862151249 x + 2776752605018538 x + 2767839862418518 x 75 74 73 - 993286665352613 x - 5771042167223380 x - 5411231377628956 x 72 71 70 + 3904707980047680 x + 6802245351932606 x + 3359939831100664 x 69 68 67 - 1683194496958176 x - 4267439489627202 x - 3307435425636736 x 66 65 64 - 382039250033604 x + 2069423994401280 x + 2534355312897568 x 63 62 61 + 1163804458882141 x - 679228904117286 x - 1516562296170256 x 60 59 58 - 815846465157237 x + 437359389160088 x + 292461778516936 x 57 56 55 + 178718877104816 x + 67265912176232 x - 12664884459768 x 54 53 52 - 46953884952092 x - 43702145284596 x - 22003894972024 x 51 50 49 - 106842590216 x + 11338338408060 x + 10666316802008 x 48 47 46 + 3052661326734 x - 3442002177908 x - 3081242720164 x 45 44 43 + 1691719313234 x - 222708589712 x - 104612127480 x 42 41 40 39 - 36324596712 x + 2812296044 x + 18862134000 x + 18393692140 x 38 37 36 35 + 9477758512 x - 356477352 x - 5940686872 x - 5525543412 x 34 33 32 31 - 1073482900 x + 2741345838 x + 1846215616 x - 1891603498 x 30 29 28 27 26 + 402270268 x - 4656000 x - 704680 x + 1459424 x + 2377760 x 25 24 23 22 21 + 2221568 x + 1308880 x + 171720 x - 596384 x - 629648 x 20 19 18 17 16 - 69564 x + 428634 x + 224305 x - 355750 x + 133438 x 15 14 13 12 11 10 9 - 15277 x + 324 x - 156 x - 360 x - 330 x - 152 x + 52 x 8 7 6 5 4 3 2 + 162 x + 120 x - 20 x - 108 x - 28 x + 101 x - 54 x + 12 x - 1) / 14 13 12 11 10 9 / ((16384 x + 8192 x + 4096 x + 2048 x + 1024 x + 512 x / 8 7 6 5 4 3 2 91 + 256 x + 128 x + 64 x + 32 x + 16 x + 8 x + 4 x + 2 x - 1) (x 90 89 88 87 86 85 + 21 x + 308 x + 4200 x + 56840 x + 769692 x + 10425688 x 84 83 82 81 - 69605246 x + 15473686 x + 356097996 x + 235374252 x 80 79 78 77 - 1491428704 x - 3418867016 x + 2390997876 x + 22425844562 x 76 75 74 73 + 25595945908 x - 78298915422 x - 279171714716 x - 72963349864 x 72 71 70 + 1244986639112 x + 355366745556 x - 1450213417192 x 69 68 67 - 1880160014814 x + 110721653710 x + 2890745401436 x 66 65 64 + 3157196655468 x - 619377462064 x - 5388740286496 x 63 62 61 - 5258115071576 x + 1953919891138 x + 9259691257865 x 60 59 58 + 5323432522759 x - 7817452326392 x - 4693024328992 x 57 56 55 - 1089850235392 x + 1978883601440 x + 2926974637280 x 54 53 52 + 1687655200252 x - 383456210484 x - 1655335584696 x 51 50 49 - 1375108676648 x - 85235367408 x + 926568851752 x 48 47 46 + 774133707608 x - 146801472116 x - 505406296688 x 45 44 43 42 + 144684548972 x - 47667449848 x - 8305444288 x + 12764787072 x 41 40 39 38 + 17264555264 x + 10518355904 x + 583143772 x - 5603553332 x 37 36 35 34 - 5426154776 x - 1173456616 x + 2391848736 x + 2119563352 x 33 32 31 30 - 634940528 x - 1424879876 x + 936610019 x - 165655709 x 29 28 27 26 25 + 20804 x - 5952296 x - 6206440 x - 3127484 x + 410792 x 24 23 22 21 20 + 2217634 x + 1737310 x + 68332 x - 1040484 x - 661680 x 19 18 17 16 15 14 + 430272 x + 537556 x - 484830 x + 149276 x - 16718 x - 1260 x 13 12 11 10 9 8 7 - 1080 x - 360 x + 300 x + 520 x + 290 x - 90 x - 260 x 6 5 4 3 2 - 100 x + 144 x + 104 x - 152 x + 66 x - 13 x + 1)) and in Maple notation 2*x^2*(2202927104*x^97-2714320896*x^96-19665479680*x^95-2217411840*x^94+ 119072541792*x^93+194797171244*x^92-366352500514*x^91-1721204784326*x^90-\ 923005206840*x^89+7930254955184*x^88+18376496787552*x^87-21096967369576*x^86-\ 268978879923960*x^85-76781535909108*x^84+541451942216572*x^83+730522973671272*x ^82-124100301220312*x^81-1374453998368544*x^80-1454445181913474*x^79+ 409365862151249*x^78+2776752605018538*x^77+2767839862418518*x^76-\ 993286665352613*x^75-5771042167223380*x^74-5411231377628956*x^73+ 3904707980047680*x^72+6802245351932606*x^71+3359939831100664*x^70-\ 1683194496958176*x^69-4267439489627202*x^68-3307435425636736*x^67-\ 382039250033604*x^66+2069423994401280*x^65+2534355312897568*x^64+ 1163804458882141*x^63-679228904117286*x^62-1516562296170256*x^61-\ 815846465157237*x^60+437359389160088*x^59+292461778516936*x^58+178718877104816* x^57+67265912176232*x^56-12664884459768*x^55-46953884952092*x^54-43702145284596 *x^53-22003894972024*x^52-106842590216*x^51+11338338408060*x^50+10666316802008* x^49+3052661326734*x^48-3442002177908*x^47-3081242720164*x^46+1691719313234*x^ 45-222708589712*x^44-104612127480*x^43-36324596712*x^42+2812296044*x^41+ 18862134000*x^40+18393692140*x^39+9477758512*x^38-356477352*x^37-5940686872*x^ 36-5525543412*x^35-1073482900*x^34+2741345838*x^33+1846215616*x^32-1891603498*x ^31+402270268*x^30-4656000*x^29-704680*x^28+1459424*x^27+2377760*x^26+2221568*x ^25+1308880*x^24+171720*x^23-596384*x^22-629648*x^21-69564*x^20+428634*x^19+ 224305*x^18-355750*x^17+133438*x^16-15277*x^15+324*x^14-156*x^13-360*x^12-330*x ^11-152*x^10+52*x^9+162*x^8+120*x^7-20*x^6-108*x^5-28*x^4+101*x^3-54*x^2+12*x-1 )/(16384*x^14+8192*x^13+4096*x^12+2048*x^11+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^91+21*x^90+308*x^89+4200*x^88+56840* x^87+769692*x^86+10425688*x^85-69605246*x^84+15473686*x^83+356097996*x^82+ 235374252*x^81-1491428704*x^80-3418867016*x^79+2390997876*x^78+22425844562*x^77 +25595945908*x^76-78298915422*x^75-279171714716*x^74-72963349864*x^73+ 1244986639112*x^72+355366745556*x^71-1450213417192*x^70-1880160014814*x^69+ 110721653710*x^68+2890745401436*x^67+3157196655468*x^66-619377462064*x^65-\ 5388740286496*x^64-5258115071576*x^63+1953919891138*x^62+9259691257865*x^61+ 5323432522759*x^60-7817452326392*x^59-4693024328992*x^58-1089850235392*x^57+ 1978883601440*x^56+2926974637280*x^55+1687655200252*x^54-383456210484*x^53-\ 1655335584696*x^52-1375108676648*x^51-85235367408*x^50+926568851752*x^49+ 774133707608*x^48-146801472116*x^47-505406296688*x^46+144684548972*x^45-\ 47667449848*x^44-8305444288*x^43+12764787072*x^42+17264555264*x^41+10518355904* x^40+583143772*x^39-5603553332*x^38-5426154776*x^37-1173456616*x^36+2391848736* x^35+2119563352*x^34-634940528*x^33-1424879876*x^32+936610019*x^31-165655709*x^ 30+20804*x^29-5952296*x^28-6206440*x^27-3127484*x^26+410792*x^25+2217634*x^24+ 1737310*x^23+68332*x^22-1040484*x^21-661680*x^20+430272*x^19+537556*x^18-484830 *x^17+149276*x^16-16718*x^15-1260*x^14-1080*x^13-360*x^12+300*x^11+520*x^10+290 *x^9-90*x^8-260*x^7-100*x^6+144*x^5+104*x^4-152*x^3+66*x^2-13*x+1) Theorem number, 14, Let a(n) be the, 15, -bonacci numbers that are defined via the generating function infinity ----- \ n 15 14 13 12 11 10 9 8 7 6 ) a(n) x = x/(-x - x - x - x - x - x - x - x - x - x / ----- n = 0 5 4 3 2 - x - x - x - x - x + 1) Let b(n) be the binomial convolution n ----- \ b(n) = ) binomial(n, r) a(r) a(n - r) / ----- r = 0 Then the generating function of the sequence b(n) is infinity ----- \ n 2 105 104 ) b(n) x = 2 x (95961941051520 x - 10676256749760 x / ----- n = 0 103 102 101 - 466183033957824 x - 451802339765504 x + 1072647006365312 x 100 99 98 + 2834121999354880 x + 401646462800768 x - 8066745136744704 x 97 96 95 - 12182038052648512 x + 7265362064854272 x + 46330095353265984 x 94 93 92 + 41937337986665984 x - 95248504682496640 x - 313781909321123328 x 91 90 89 + 18219642826883584 x + 494566387515672512 x + 465044009583620800 x 88 87 86 - 141892701361231360 x - 734611603268214016 x - 660733014769529408 x 85 84 83 + 114349317940975040 x + 905974876467638464 x + 913312661981582912 x 82 81 80 + 1435622376023264 x - 1061671143961275360 x - 1200781208341490400 x 79 78 77 - 68780407370181280 x + 1244189231990425408 x + 759655251701842112 x 76 75 74 - 14271631102902784 x - 458458705914440704 x - 456600050590458048 x 73 72 71 - 187667353245483200 x + 92792027372133376 x + 218849097535809664 x 70 69 68 + 171553908732912160 x + 38261287077464288 x - 71259006094460608 x 67 66 65 - 94850689515109136 x - 43028594661532736 x + 21859270670350688 x 64 63 62 + 37025479456292416 x + 1573315808974000 x + 819121445588736 x 61 60 59 - 1157455730526368 x - 1955065689738384 x - 1616151268681616 x 58 57 56 - 789542170350944 x - 41417793628800 x + 350902659215312 x 55 54 53 + 379754139433264 x + 196393607155680 x - 11139464515712 x 52 51 50 - 110603827840616 x - 75152325387016 x + 16116952186448 x 49 48 47 + 42748469867168 x - 19392256136552 x + 512769686984 x 46 45 44 + 492138012560 x + 481464154080 x + 353715933504 x 43 42 41 40 + 171903415280 x + 16434398720 x - 67635618080 x - 76788762752 x 39 38 37 36 - 37706740688 x + 9824040256 x + 31824722160 x + 17959691420 x 35 34 33 32 - 10008580300 x - 14367793032 x + 9683648060 x - 1775812952 x 31 30 29 28 27 + 391276 x + 5761144 x + 6487764 x + 3188536 x - 1288348 x 26 25 24 23 22 - 4318284 x - 4557172 x - 2356068 x + 503524 x + 1893408 x 21 20 19 18 17 + 944220 x - 894036 x - 926912 x + 947930 x - 296438 x 16 15 14 13 12 11 10 + 31142 x - 1542 x - 1086 x - 450 x + 114 x + 414 x + 388 x 9 8 7 6 5 4 3 2 + 132 x - 140 x - 218 x - 58 x + 138 x + 86 x - 144 x + 65 x / 15 14 13 12 11 - 13 x + 1) / ((32768 x + 16384 x + 8192 x + 4096 x + 2048 x / 10 9 8 7 6 5 4 3 + 1024 x + 512 x + 256 x + 128 x + 64 x + 32 x + 16 x + 8 x 2 105 104 103 102 + 4 x + 2 x - 1) (128 x - 2688 x + 17920 x - 12544 x 101 100 99 98 97 - 232960 x - 155904 x + 3234560 x + 13959808 x - 12252416 x 96 95 94 93 - 351524480 x - 1142172672 x + 3345487104 x + 39505594368 x 92 91 90 + 83253330944 x - 565626567680 x - 165458174656 x 89 88 87 + 1563202470784 x + 1866703297344 x - 2361355828736 x 86 85 84 - 8143142379264 x - 3644834479104 x + 18557406705280 x 83 82 81 + 35284098389120 x - 5779241574144 x - 108801105962368 x 80 79 78 - 128198383672192 x + 134956106533120 x + 457187238945024 x 77 76 75 - 75695087154048 x - 462910431150656 x - 353589047975648 x 74 73 72 + 130257487506272 x + 527177202838624 x + 432709738075936 x 71 70 69 - 110943057746240 x - 605495242188608 x - 535398639095104 x 68 67 66 + 99256910044960 x + 678614540135616 x + 527110303201792 x 65 64 63 - 249668857859072 x - 625880007646688 x - 4745660137216 x 62 61 60 + 82784743183872 x + 132191532259040 x + 95422214392240 x 59 58 57 + 16181098690880 x - 45850447753568 x - 58488168002528 x 56 55 54 - 28898653971760 x + 10555898595840 x + 28852967582688 x 53 52 51 + 17882057158400 x - 4670583117984 x - 13673908628416 x 50 49 48 - 2896976536384 x + 6658434705600 x - 2428174226464 x 47 46 45 44 + 363402153120 x + 315724862208 x + 185301415400 x + 33975073800 x 43 42 41 40 - 65863227184 x - 84974651328 x - 44141962824 x + 8741925000 x 39 38 37 36 + 32859681568 x + 19568479240 x - 6995655576 x - 15367163936 x 35 34 33 32 - 185359184 x + 9887507688 x - 4835853968 x + 742744568 x 31 30 29 28 27 - 39464288 x - 19168228 x + 1569736 x + 13214772 x + 12857040 x 26 25 24 23 22 + 4838836 x - 3204872 x - 5671236 x - 2413480 x + 1934036 x 21 20 19 18 17 + 2532672 x - 495604 x - 1779408 x + 1237128 x - 333672 x 16 15 14 13 12 11 10 + 28556 x - 1542 x + 630 x + 1530 x + 1170 x + 210 x - 530 x 9 8 7 6 5 4 3 2 - 590 x - 120 x + 310 x + 250 x - 138 x - 202 x + 206 x - 78 x + 14 x - 1)) and in Maple notation 2*x^2*(95961941051520*x^105-10676256749760*x^104-466183033957824*x^103-\ 451802339765504*x^102+1072647006365312*x^101+2834121999354880*x^100+ 401646462800768*x^99-8066745136744704*x^98-12182038052648512*x^97+ 7265362064854272*x^96+46330095353265984*x^95+41937337986665984*x^94-\ 95248504682496640*x^93-313781909321123328*x^92+18219642826883584*x^91+ 494566387515672512*x^90+465044009583620800*x^89-141892701361231360*x^88-\ 734611603268214016*x^87-660733014769529408*x^86+114349317940975040*x^85+ 905974876467638464*x^84+913312661981582912*x^83+1435622376023264*x^82-\ 1061671143961275360*x^81-1200781208341490400*x^80-68780407370181280*x^79+ 1244189231990425408*x^78+759655251701842112*x^77-14271631102902784*x^76-\ 458458705914440704*x^75-456600050590458048*x^74-187667353245483200*x^73+ 92792027372133376*x^72+218849097535809664*x^71+171553908732912160*x^70+ 38261287077464288*x^69-71259006094460608*x^68-94850689515109136*x^67-\ 43028594661532736*x^66+21859270670350688*x^65+37025479456292416*x^64+ 1573315808974000*x^63+819121445588736*x^62-1157455730526368*x^61-\ 1955065689738384*x^60-1616151268681616*x^59-789542170350944*x^58-41417793628800 *x^57+350902659215312*x^56+379754139433264*x^55+196393607155680*x^54-\ 11139464515712*x^53-110603827840616*x^52-75152325387016*x^51+16116952186448*x^ 50+42748469867168*x^49-19392256136552*x^48+512769686984*x^47+492138012560*x^46+ 481464154080*x^45+353715933504*x^44+171903415280*x^43+16434398720*x^42-\ 67635618080*x^41-76788762752*x^40-37706740688*x^39+9824040256*x^38+31824722160* x^37+17959691420*x^36-10008580300*x^35-14367793032*x^34+9683648060*x^33-\ 1775812952*x^32+391276*x^31+5761144*x^30+6487764*x^29+3188536*x^28-1288348*x^27 -4318284*x^26-4557172*x^25-2356068*x^24+503524*x^23+1893408*x^22+944220*x^21-\ 894036*x^20-926912*x^19+947930*x^18-296438*x^17+31142*x^16-1542*x^15-1086*x^14-\ 450*x^13+114*x^12+414*x^11+388*x^10+132*x^9-140*x^8-218*x^7-58*x^6+138*x^5+86*x ^4-144*x^3+65*x^2-13*x+1)/(32768*x^15+16384*x^14+8192*x^13+4096*x^12+2048*x^11+ 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)/(128* x^105-2688*x^104+17920*x^103-12544*x^102-232960*x^101-155904*x^100+3234560*x^99 +13959808*x^98-12252416*x^97-351524480*x^96-1142172672*x^95+3345487104*x^94+ 39505594368*x^93+83253330944*x^92-565626567680*x^91-165458174656*x^90+ 1563202470784*x^89+1866703297344*x^88-2361355828736*x^87-8143142379264*x^86-\ 3644834479104*x^85+18557406705280*x^84+35284098389120*x^83-5779241574144*x^82-\ 108801105962368*x^81-128198383672192*x^80+134956106533120*x^79+457187238945024* x^78-75695087154048*x^77-462910431150656*x^76-353589047975648*x^75+ 130257487506272*x^74+527177202838624*x^73+432709738075936*x^72-110943057746240* x^71-605495242188608*x^70-535398639095104*x^69+99256910044960*x^68+ 678614540135616*x^67+527110303201792*x^66-249668857859072*x^65-625880007646688* x^64-4745660137216*x^63+82784743183872*x^62+132191532259040*x^61+95422214392240 *x^60+16181098690880*x^59-45850447753568*x^58-58488168002528*x^57-\ 28898653971760*x^56+10555898595840*x^55+28852967582688*x^54+17882057158400*x^53 -4670583117984*x^52-13673908628416*x^51-2896976536384*x^50+6658434705600*x^49-\ 2428174226464*x^48+363402153120*x^47+315724862208*x^46+185301415400*x^45+ 33975073800*x^44-65863227184*x^43-84974651328*x^42-44141962824*x^41+8741925000* x^40+32859681568*x^39+19568479240*x^38-6995655576*x^37-15367163936*x^36-\ 185359184*x^35+9887507688*x^34-4835853968*x^33+742744568*x^32-39464288*x^31-\ 19168228*x^30+1569736*x^29+13214772*x^28+12857040*x^27+4838836*x^26-3204872*x^ 25-5671236*x^24-2413480*x^23+1934036*x^22+2532672*x^21-495604*x^20-1779408*x^19 +1237128*x^18-333672*x^17+28556*x^16-1542*x^15+630*x^14+1530*x^13+1170*x^12+210 *x^11-530*x^10-590*x^9-120*x^8+310*x^7+250*x^6-138*x^5-202*x^4+206*x^3-78*x^2+ 14*x-1) Theorem number, 15, Let a(n) be the, 16, -bonacci numbers that are defined via the generating function infinity ----- \ n 16 15 14 13 12 11 10 9 8 7 ) a(n) x = x/(-x - x - x - x - x - x - x - x - x - x / ----- n = 0 6 5 4 3 2 - x - x - x - x - x - x + 1) Let b(n) be the binomial convolution n ----- \ b(n) = ) binomial(n, r) a(r) a(n - r) / ----- r = 0 Then the generating function of the sequence b(n) is infinity ----- \ n 2 127 126 ) b(n) x = 2 x (274877906944 x - 751619276800 x / ----- n = 0 125 124 123 - 2746564476928 x + 4439386619904 x + 26460956672000 x 122 121 120 + 5615921578240 x - 181947525797052 x - 354088415855490 x 119 118 117 + 662019742914832 x + 3728260971262608 x + 2210788984990328 x 116 115 114 - 22192720127117928 x - 55040718182614288 x + 87541779042310928 x 113 112 + 1068285334047748540 x + 1210029607644172 x 111 110 - 3237575528786550672 x - 3320876878586219088 x 109 108 + 3707701350507349064 x + 11522417980631722112 x 107 106 + 6220711748427777812 x - 15591908550225877334 x 105 104 - 31512960358868857725 x - 8541060776433017959 x 103 102 + 50714086580045129632 x + 79763775323018655188 x 101 100 - 1345752990426060920 x - 165605349263759780548 x 99 98 - 202873308643759775224 x + 129924983711088881006 x 97 96 + 320191077929926459156 x + 165850994310241756470 x 95 94 - 147883781457987721264 x - 316370888948796962172 x 93 92 - 200757516566559938104 x + 79191341347040189040 x 91 90 + 276069239223885105676 x + 231980539821536385451 x 89 88 + 82164232934702554 x - 214456905659233883021 x 87 86 - 231931713366921342512 x - 48354995065865447368 x 85 84 + 157740252819433460928 x + 177677421520190666576 x 83 82 + 3808464749757642536 x - 58293035301647199064 x 81 80 - 57696459544021457424 x - 27961298983715545168 x 79 78 + 2742916956152633136 x + 18628750391381285184 x 77 76 + 17970805280235186704 x + 8055579856324188614 x 75 74 - 2049962818692413721 x - 6753383406959473267 x 73 72 - 5522351164799903747 x - 1392891245898141095 x 71 70 + 1867643001927590034 x + 2240045499130161318 x 69 68 67 + 496437446287394592 x - 733513921014651010 x + 14142715482200440 x 66 65 64 - 66155207753953380 x - 56696664492211944 x - 25489788577468486 x 63 62 61 + 279306504040152 x + 12575950074792508 x + 13088106633864572 x 60 59 58 + 7369445268657911 x + 942067731214196 x - 2796823417942150 x 57 56 55 - 3091961261164496 x - 1231974376414731 x + 666415761961100 x 54 53 52 + 1064565137447038 x + 86456651855928 x - 573572889751928 x 51 50 49 + 184724075264992 x + 2647590803552 x + 3182219309540 x 48 47 46 + 1158128883788 x - 667206311668 x - 1482145402934 x 45 44 43 - 1360751820041 x - 732202863579 x - 63834441030 x 42 41 40 + 323199346580 x + 336016515485 x + 102675835719 x 39 38 37 36 - 124813813642 x - 145314850906 x + 14358148743 x + 94755176391 x 35 34 33 32 - 48282600640 x + 7669648678 x + 127567180 x + 70931362 x 31 30 29 28 27 + 19785060 x - 10332211 x - 18492910 x - 11339309 x + 1147056 x 26 25 24 23 22 + 9673019 x + 9495614 x + 2557327 x - 4099564 x - 4128991 x 21 20 19 18 17 + 1317418 x + 3227411 x - 2424832 x + 652112 x - 68032 x 16 15 14 13 12 11 10 - 690 x + 771 x + 1401 x + 1215 x + 537 x - 177 x - 539 x 9 8 7 6 5 4 3 2 - 415 x - 3 x + 293 x + 191 x - 143 x - 177 x + 197 x - 77 x / 16 15 14 13 12 + 14 x - 1) / ((65536 x + 32768 x + 16384 x + 8192 x + 4096 x / 11 10 9 8 7 6 5 4 + 2048 x + 1024 x + 512 x + 256 x + 128 x + 64 x + 32 x + 16 x 3 2 120 119 118 117 + 8 x + 4 x + 2 x - 1) (x + 24 x + 400 x + 6188 x 116 115 114 113 + 94996 x + 1459280 x + 22423204 x + 344559638 x 112 111 110 109 - 3295392406 x + 4047014984 x + 19588744624 x - 12554962212 x 108 107 106 - 133322044468 x - 113518260772 x + 689735111598 x 105 104 103 + 1976856119657 x - 1175547844237 x - 15963292877688 x 102 101 100 - 21430395742968 x + 72587206887296 x + 288365483881416 x 99 98 97 + 41534885714616 x - 1748140647790060 x - 271695992705728 x 96 95 94 + 3144333563616084 x + 3333606520791600 x - 2425953281865224 x 93 92 91 - 8963126621203536 x - 5803035233509620 x + 10047856717650888 x 90 89 88 + 22959447468346897 x + 8643884437701194 x - 32668683795850211 x 87 86 85 - 54466262236679404 x - 2453553312256738 x + 92653656381444676 x 84 83 82 + 86319252221141176 x - 82675164411408830 x - 97497592988260538 x 81 80 79 - 30041032945746402 x + 50531719403400662 x + 79598663910529736 x 78 77 76 + 39913495864905816 x - 29068450886412596 x - 67362141075331962 x 75 74 73 - 45039094167429455 x + 14038950600825491 x + 54409155557520179 x 72 71 70 + 39684225530368803 x - 11361143047858526 x - 40990028572958658 x 69 68 67 - 17002882129395048 x + 18260201519939592 x + 4172631817262464 x 66 65 64 + 4304195160234800 x + 1280850051610640 x - 1366495957109088 x 63 62 61 - 2240505263267536 x - 1514223794102112 x - 184185589480352 x 60 59 58 + 744490148068345 x + 828118067058588 x + 300742757618190 x 57 56 55 - 244206918125256 x - 361057732623057 x - 88089444651844 x 54 53 52 + 157614588837830 x + 78458522688090 x - 95735561041187 x 51 50 49 + 31443267504772 x + 1739410117128 x + 147815604906 x 48 47 46 - 1088491844938 x - 1341558107904 x - 823338316618 x 45 44 43 42 - 87952272977 x + 376440416269 x + 394241278534 x + 122190685552 x 41 40 39 38 - 131612675301 x - 161455201771 x - 13565090838 x + 92416177820 x 37 36 35 34 + 27000395775 x - 61614589911 x + 24193673316 x - 3605306044 x 33 32 31 30 29 - 9658368 x + 71869848 x + 81486384 x + 41467629 x - 6871794 x 28 27 26 25 24 - 31696229 x - 25492872 x - 3168083 x + 12987906 x + 11223469 x 23 22 21 20 19 - 1391714 x - 7691589 x - 669004 x + 5438833 x - 3085530 x 18 17 16 15 14 13 + 718499 x - 62442 x + 5316 x + 4203 x + 1401 x - 1029 x 12 11 10 9 8 7 6 - 1839 x - 1089 x + 181 x + 831 x + 481 x - 259 x - 449 x 5 4 3 2 + 77 x + 343 x - 271 x + 91 x - 15 x + 1)) and in Maple notation 2*x^2*(274877906944*x^127-751619276800*x^126-2746564476928*x^125+4439386619904* x^124+26460956672000*x^123+5615921578240*x^122-181947525797052*x^121-\ 354088415855490*x^120+662019742914832*x^119+3728260971262608*x^118+ 2210788984990328*x^117-22192720127117928*x^116-55040718182614288*x^115+ 87541779042310928*x^114+1068285334047748540*x^113+1210029607644172*x^112-\ 3237575528786550672*x^111-3320876878586219088*x^110+3707701350507349064*x^109+ 11522417980631722112*x^108+6220711748427777812*x^107-15591908550225877334*x^106 -31512960358868857725*x^105-8541060776433017959*x^104+50714086580045129632*x^ 103+79763775323018655188*x^102-1345752990426060920*x^101-165605349263759780548* x^100-202873308643759775224*x^99+129924983711088881006*x^98+ 320191077929926459156*x^97+165850994310241756470*x^96-147883781457987721264*x^ 95-316370888948796962172*x^94-200757516566559938104*x^93+79191341347040189040*x ^92+276069239223885105676*x^91+231980539821536385451*x^90+82164232934702554*x^ 89-214456905659233883021*x^88-231931713366921342512*x^87-48354995065865447368*x ^86+157740252819433460928*x^85+177677421520190666576*x^84+3808464749757642536*x ^83-58293035301647199064*x^82-57696459544021457424*x^81-27961298983715545168*x^ 80+2742916956152633136*x^79+18628750391381285184*x^78+17970805280235186704*x^77 +8055579856324188614*x^76-2049962818692413721*x^75-6753383406959473267*x^74-\ 5522351164799903747*x^73-1392891245898141095*x^72+1867643001927590034*x^71+ 2240045499130161318*x^70+496437446287394592*x^69-733513921014651010*x^68+ 14142715482200440*x^67-66155207753953380*x^66-56696664492211944*x^65-\ 25489788577468486*x^64+279306504040152*x^63+12575950074792508*x^62+ 13088106633864572*x^61+7369445268657911*x^60+942067731214196*x^59-\ 2796823417942150*x^58-3091961261164496*x^57-1231974376414731*x^56+ 666415761961100*x^55+1064565137447038*x^54+86456651855928*x^53-573572889751928* x^52+184724075264992*x^51+2647590803552*x^50+3182219309540*x^49+1158128883788*x ^48-667206311668*x^47-1482145402934*x^46-1360751820041*x^45-732202863579*x^44-\ 63834441030*x^43+323199346580*x^42+336016515485*x^41+102675835719*x^40-\ 124813813642*x^39-145314850906*x^38+14358148743*x^37+94755176391*x^36-\ 48282600640*x^35+7669648678*x^34+127567180*x^33+70931362*x^32+19785060*x^31-\ 10332211*x^30-18492910*x^29-11339309*x^28+1147056*x^27+9673019*x^26+9495614*x^ 25+2557327*x^24-4099564*x^23-4128991*x^22+1317418*x^21+3227411*x^20-2424832*x^ 19+652112*x^18-68032*x^17-690*x^16+771*x^15+1401*x^14+1215*x^13+537*x^12-177*x^ 11-539*x^10-415*x^9-3*x^8+293*x^7+191*x^6-143*x^5-177*x^4+197*x^3-77*x^2+14*x-1 )/(65536*x^16+32768*x^15+16384*x^14+8192*x^13+4096*x^12+2048*x^11+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^120+24*x^119+ 400*x^118+6188*x^117+94996*x^116+1459280*x^115+22423204*x^114+344559638*x^113-\ 3295392406*x^112+4047014984*x^111+19588744624*x^110-12554962212*x^109-\ 133322044468*x^108-113518260772*x^107+689735111598*x^106+1976856119657*x^105-\ 1175547844237*x^104-15963292877688*x^103-21430395742968*x^102+72587206887296*x^ 101+288365483881416*x^100+41534885714616*x^99-1748140647790060*x^98-\ 271695992705728*x^97+3144333563616084*x^96+3333606520791600*x^95-\ 2425953281865224*x^94-8963126621203536*x^93-5803035233509620*x^92+ 10047856717650888*x^91+22959447468346897*x^90+8643884437701194*x^89-\ 32668683795850211*x^88-54466262236679404*x^87-2453553312256738*x^86+ 92653656381444676*x^85+86319252221141176*x^84-82675164411408830*x^83-\ 97497592988260538*x^82-30041032945746402*x^81+50531719403400662*x^80+ 79598663910529736*x^79+39913495864905816*x^78-29068450886412596*x^77-\ 67362141075331962*x^76-45039094167429455*x^75+14038950600825491*x^74+ 54409155557520179*x^73+39684225530368803*x^72-11361143047858526*x^71-\ 40990028572958658*x^70-17002882129395048*x^69+18260201519939592*x^68+ 4172631817262464*x^67+4304195160234800*x^66+1280850051610640*x^65-\ 1366495957109088*x^64-2240505263267536*x^63-1514223794102112*x^62-\ 184185589480352*x^61+744490148068345*x^60+828118067058588*x^59+300742757618190* x^58-244206918125256*x^57-361057732623057*x^56-88089444651844*x^55+ 157614588837830*x^54+78458522688090*x^53-95735561041187*x^52+31443267504772*x^ 51+1739410117128*x^50+147815604906*x^49-1088491844938*x^48-1341558107904*x^47-\ 823338316618*x^46-87952272977*x^45+376440416269*x^44+394241278534*x^43+ 122190685552*x^42-131612675301*x^41-161455201771*x^40-13565090838*x^39+ 92416177820*x^38+27000395775*x^37-61614589911*x^36+24193673316*x^35-3605306044* x^34-9658368*x^33+71869848*x^32+81486384*x^31+41467629*x^30-6871794*x^29-\ 31696229*x^28-25492872*x^27-3168083*x^26+12987906*x^25+11223469*x^24-1391714*x^ 23-7691589*x^22-669004*x^21+5438833*x^20-3085530*x^19+718499*x^18-62442*x^17+ 5316*x^16+4203*x^15+1401*x^14-1029*x^13-1839*x^12-1089*x^11+181*x^10+831*x^9+ 481*x^8-259*x^7-449*x^6+77*x^5+343*x^4-271*x^3+91*x^2-15*x+1) Theorem number, 16, Let a(n) be the, 17, -bonacci numbers that are defined via the generating function infinity ----- \ n 17 16 15 14 13 12 11 10 9 8 ) a(n) x = x/(-x - x - x - x - x - x - x - x - x - x / ----- n = 0 7 6 5 4 3 2 - x - x - x - x - x - x - x + 1) Let b(n) be the binomial convolution n ----- \ b(n) = ) binomial(n, r) a(r) a(n - r) / ----- r = 0 Then the generating function of the sequence b(n) is infinity ----- \ n 2 136 135 ) b(n) x = - 2 x (100699895579603200 x - 83848652269150208 x / ----- n = 0 134 133 - 648913044999478016 x - 167331030747139584 x 132 131 + 2635496783749098208 x + 3920346948553016320 x 130 129 - 5179551674311147312 x - 21942956577473183456 x 128 127 - 10874385221696878135 x + 66286432096347641296 x 126 125 + 133551192472017796068 x - 55683580739460787712 x 124 123 - 576032243464102411932 x - 608648367889456588200 x 122 121 + 1515709493852623104382 x + 5439369505167121408704 x 120 119 - 1206290771076053394574 x - 12111365402000894627760 x 118 117 - 9651410947303023928764 x + 9849034725392313389248 x 116 115 + 26148214529260496024748 x + 14719995625790735686900 x 114 113 - 21936623264143018972847 x - 47051948699515665654496 x 112 111 - 23272835571776688026749 x + 40184907801501456914520 x 110 109 + 80126238601979503397424 x + 35911609443934473243392 x 108 107 - 73278015547252789622788 x - 135078529860067357589728 x 106 105 - 43000837611857161106992 x + 140769181905252385854992 x 104 103 + 132892889458638613321340 x + 8831676154214716794016 x 102 101 - 94710661365614791868380 x - 103231270374797370482552 x 100 99 - 33452168576862505860123 x + 45751835279663345865656 x 98 97 + 75402428384562746864086 x + 45205446288871079731016 x 96 95 - 10475200734820775449751 x - 46430182562463732405692 x 94 93 - 40792254408805693545556 x - 6378664877241300231978 x 92 91 + 24484592086101749312120 x + 27504292602398034782704 x 90 89 + 6027818407891931678362 x - 12450917647073463748416 x 88 87 - 7270759696588902834150 x - 3037944555994013957480 x 86 85 + 229041645947866484285 x + 1791469264291293873028 x 84 83 + 1773743471899592933433 x + 919494971514717490764 x 82 81 + 16505057707466528875 x - 475335040176121668556 x 80 79 - 494347782944464938651 x - 239275996246552432734 x 78 77 + 32458195121391796332 x + 157425344268509916466 x 76 75 + 119894449464582418910 x + 13678559707409886880 x 74 73 - 48428119268891556306 x - 26218169836595659544 x 72 71 + 15531435855880875897 x - 3413729584972883648 x 70 69 68 - 926730131442534568 x + 187188879922286368 x + 548483488445892758 x 67 66 65 + 489962254448225676 x + 263192743573146948 x + 43802091388302158 x 64 63 62 - 81123123069313671 x - 102345801193814928 x - 58674959517712066 x 61 60 59 - 2392842922406954 x + 28346446742079408 x + 23788959201517866 x 58 57 56 + 1090080871327360 x - 12412208028958717 x - 4525457380509896 x 55 54 53 + 6896344037450472 x - 1588932423968098 x + 47118958063840 x 52 51 50 - 2324184993610 x - 18240329686196 x - 16558803296619 x 49 48 47 - 7896406301944 x + 582229631909 x + 5203424383938 x 46 45 44 + 5412965378418 x + 2790802950980 x - 255203931230 x 43 42 41 - 1788485976879 x - 1287608614296 x + 182993979925 x 40 39 38 + 893542668882 x + 169523697202 x - 571645269914 x 37 36 35 34 + 235272667866 x - 31858995333 x - 339184680 x - 359508152 x 33 32 31 30 - 274816572 x - 151121383 x - 37929010 x + 32236230 x 29 28 27 26 25 + 49266169 x + 26346212 x - 7942216 x - 25602572 x - 16416139 x 24 23 22 21 20 + 5267470 x + 13645036 x + 133204 x - 10141261 x + 6011886 x 19 18 17 16 15 14 - 1445034 x + 134317 x + 4896 x + 3777 x + 1545 x - 486 x 13 12 11 10 9 8 7 - 1515 x - 1374 x - 483 x + 418 x + 709 x + 300 x - 289 x 6 5 4 3 2 / 17 - 378 x + 99 x + 310 x - 261 x + 90 x - 15 x + 1) / ((131072 x / 16 15 14 13 12 11 + 65536 x + 32768 x + 16384 x + 8192 x + 4096 x + 2048 x 10 9 8 7 6 5 4 3 + 1024 x + 512 x + 256 x + 128 x + 64 x + 32 x + 16 x + 8 x 2 136 135 134 133 + 4 x + 2 x - 1) (256 x - 7168 x + 71424 x - 223488 x 132 131 130 129 - 687264 x + 2884800 x + 17333808 x + 4345488 x 128 127 126 125 - 388962159 x - 1859650800 x + 2491856260 x + 64310075200 x 124 123 122 + 214950458948 x - 902249671496 x - 10222584720082 x 121 120 119 - 19260382265696 x + 210352945984898 x - 43569187076976 x 118 117 116 - 802329257477276 x - 458977271284672 x + 2337194820533676 x 115 114 113 + 4376853269397444 x - 2718008523518791 x - 19311683347832464 x 112 111 - 16645885022815245 x + 46701550876578112 x 110 109 + 125708561579244112 x + 7620043145683424 x 108 107 - 442701787654418500 x - 638503809157450496 x 106 105 + 688218628330895328 x + 2758699315769864192 x 104 103 - 551831104272896372 x - 3978737056577520912 x 102 101 - 2816475222652782844 x + 2587686072406843288 x 100 99 + 6646614150613218477 x + 3716827202836342680 x 98 97 - 5101018089403286882 x - 10932167113640061392 x 96 95 - 5254872791291815823 x + 9235502989348175240 x 94 93 + 17461689702884481982 x + 6327250254578631968 x 92 91 - 16667533558411037396 x - 24049758218528605208 x 90 89 + 58675069242971794 x + 26157540077166429568 x 88 87 + 9696941979632761846 x - 3609959493113444436 x 86 85 - 10589423244117711663 x - 8596356621631299688 x 84 83 - 1072259766366174407 x + 5488309405702120428 x 82 81 + 6631268407077182019 x + 2673286226734155056 x 80 79 - 2400212848312665689 x - 4482118275167512504 x 78 77 - 2508009481563532482 x + 1066480519808500304 x 76 75 + 2739691871696251978 x + 1323433467261099336 x 74 73 - 933923090121181230 x - 1094018894567315168 x 72 71 70 + 283651653747267921 x - 102505322645009856 x + 46391525774008136 x 69 68 67 + 89149045152540400 x + 65395129544930694 x + 17822696476918936 x 66 65 64 - 19027667962947426 x - 29636760881304720 x - 18235825180210631 x 63 62 61 - 141653861583616 x + 10383084900453680 x + 8759860196277888 x 60 59 58 + 752125519227828 x - 4372108532899948 x - 2631877815662519 x 57 56 55 + 1518011960544832 x + 1523497682990148 x - 1331729528614168 x 54 53 52 + 280116798161358 x + 10716635533840 x + 18302436079102 x 51 50 49 + 12052684979860 x + 2109645599461 x - 4748600388080 x 48 47 46 - 6069113639221 x - 3258464770032 x + 405284327772 x 45 44 43 + 2206489063480 x + 1550843698059 x - 143083434276 x 42 41 40 - 992651875909 x - 412676323136 x + 457811703916 x 39 38 37 36 + 296899083568 x - 358019240148 x + 120710440968 x - 15744726273 x 35 34 33 32 - 699549456 x - 378443676 x - 24983856 x + 176124921 x 31 30 29 28 + 185095692 x + 75516693 x - 38356656 x - 77978986 x 27 26 25 24 23 - 40446932 x + 17377495 x + 36067040 x + 7961476 x - 20045188 x 22 21 20 19 18 - 7333595 x + 15763472 x - 7472166 x + 1557180 x - 144313 x 17 16 15 14 13 12 - 4896 x + 2658 x + 5748 x + 4347 x + 888 x - 1761 x 11 10 9 8 7 6 5 4 - 2100 x - 649 x + 832 x + 963 x - 36 x - 675 x - 72 x + 537 x 3 2 - 348 x + 105 x - 16 x + 1)) and in Maple notation -2*x^2*(100699895579603200*x^136-83848652269150208*x^135-648913044999478016*x^ 134-167331030747139584*x^133+2635496783749098208*x^132+3920346948553016320*x^ 131-5179551674311147312*x^130-21942956577473183456*x^129-10874385221696878135*x ^128+66286432096347641296*x^127+133551192472017796068*x^126-\ 55683580739460787712*x^125-576032243464102411932*x^124-608648367889456588200*x^ 123+1515709493852623104382*x^122+5439369505167121408704*x^121-\ 1206290771076053394574*x^120-12111365402000894627760*x^119-\ 9651410947303023928764*x^118+9849034725392313389248*x^117+ 26148214529260496024748*x^116+14719995625790735686900*x^115-\ 21936623264143018972847*x^114-47051948699515665654496*x^113-\ 23272835571776688026749*x^112+40184907801501456914520*x^111+ 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283651653747267921*x^72-102505322645009856*x^71+46391525774008136*x^70+ 89149045152540400*x^69+65395129544930694*x^68+17822696476918936*x^67-\ 19027667962947426*x^66-29636760881304720*x^65-18235825180210631*x^64-\ 141653861583616*x^63+10383084900453680*x^62+8759860196277888*x^61+ 752125519227828*x^60-4372108532899948*x^59-2631877815662519*x^58+ 1518011960544832*x^57+1523497682990148*x^56-1331729528614168*x^55+ 280116798161358*x^54+10716635533840*x^53+18302436079102*x^52+12052684979860*x^ 51+2109645599461*x^50-4748600388080*x^49-6069113639221*x^48-3258464770032*x^47+ 405284327772*x^46+2206489063480*x^45+1550843698059*x^44-143083434276*x^43-\ 992651875909*x^42-412676323136*x^41+457811703916*x^40+296899083568*x^39-\ 358019240148*x^38+120710440968*x^37-15744726273*x^36-699549456*x^35-378443676*x ^34-24983856*x^33+176124921*x^32+185095692*x^31+75516693*x^30-38356656*x^29-\ 77978986*x^28-40446932*x^27+17377495*x^26+36067040*x^25+7961476*x^24-20045188*x ^23-7333595*x^22+15763472*x^21-7472166*x^20+1557180*x^19-144313*x^18-4896*x^17+ 2658*x^16+5748*x^15+4347*x^14+888*x^13-1761*x^12-2100*x^11-649*x^10+832*x^9+963 *x^8-36*x^7-675*x^6-72*x^5+537*x^4-348*x^3+105*x^2-16*x+1) Theorem number, 17, Let a(n) be the, 18, -bonacci numbers that are defined via the generating function infinity ----- \ n 18 17 16 15 14 13 12 11 10 9 ) a(n) x = x/(-x - x - x - x - x - x - x - x - x - x / ----- n = 0 8 7 6 5 4 3 2 - x - x - x - x - x - x - x - x + 1) Let b(n) be the binomial convolution n ----- \ b(n) = ) binomial(n, r) a(r) a(n - r) / ----- r = 0 Then the generating function of the sequence b(n) is infinity ----- \ n 2 161 160 ) b(n) x = 2 x (40122452017152 x - 183337315467264 x / ----- n = 0 159 158 157 - 327160173035520 x + 1786285910900736 x + 4681921845012480 x 156 155 154 - 9649200097581312 x - 56190217184034912 x - 13604035158778564 x 153 152 + 467000173338079614 x + 1008791088295263502 x 151 150 - 2156960049973503316 x - 13135260011385554978 x 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181812079763819416*x^69+248044921058766457*x^68+335487141703550745*x^67+ 162012498009421763*x^66-50592703571519961*x^65-127852214475124550*x^64-\ 58223917871498578*x^63+37235061290125345*x^62+47167635673201369*x^61-\ 10044063535596906*x^60-26308491700122257*x^59+15925129445800886*x^58-\ 2514308103176400*x^57+245140491395064*x^56+53689684164015*x^55-60051724486243*x ^54-88228498007006*x^53-56338148135985*x^52-6843350969650*x^51+25163764519843*x ^50+27349639025952*x^49+9901545271407*x^48-7422400614387*x^47-11330726287548*x^ 46-3377216196543*x^45+4651285395258*x^44+3982767557773*x^43-1757997302566*x^42-\ 2371918524450*x^41+2001095786653*x^40-588200818652*x^39+63026051985*x^38+ 424388232*x^37+1499394321*x^36+1443660187*x^35+716489248*x^34-62314583*x^33-\ 455784284*x^32-386377643*x^31-79143872*x^30+158539585*x^29+166143251*x^28+ 16866318*x^27-91992672*x^26-51362418*x^25+44882511*x^24+34140585*x^23-43623300* x^22+17762199*x^21-3428183*x^20+269858*x^19+18784*x^18+15318*x^17+5106*x^16-\ 3732*x^15-6678*x^14-4107*x^13+303*x^12+2742*x^11+1940*x^10-373*x^9-1467*x^8-432 *x^7+882*x^6+351*x^5-795*x^4+438*x^3-120*x^2+17*x-1) Theorem number, 18, Let a(n) be the, 19, -bonacci numbers that are defined via the generating function infinity ----- \ n 19 18 17 16 15 14 13 12 11 ) a(n) x = x/(-x - x - x - x - x - x - x - x - x / ----- n = 0 10 9 8 7 6 5 4 3 2 - x - x - x - x - x - x - x - x - x - x + 1) Let b(n) be the binomial convolution n ----- \ b(n) = ) binomial(n, r) a(r) a(n - r) / ----- r = 0 Then the generating function of the sequence b(n) is infinity ----- \ n 2 171 ) b(n) x = 2 x (139550672726140897792 x / ----- n = 0 170 169 - 243206640703858984704 x - 1052883382601037195008 x 168 167 + 752611835958755528960 x + 6300915308335528548992 x 166 165 + 3551890045495878752640 x - 25484796370809711937920 x 164 163 - 50780000646044219752384 x + 46277717627003703177536 x 162 161 + 296844127912116942092864 x + 218395825950556547049376 x 160 159 - 1021541039654488809677856 x - 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352931323200*x^160-715261602272*x^159-16101241427840*x^158-52476573193632*x^157 +321886718988944*x^156+3432995532104480*x^155+5084506677935952*x^154-\ 98183358785330936*x^153+85248984599655168*x^152+457535970857212376*x^151-\ 96170419470956892*x^150-2015615421678838408*x^149-1880850550517300524*x^148+ 6281232077662718498*x^147+16657290167621964488*x^146-4355811589752677682*x^145-\ 79946396884029977275*x^144-96017416744699146336*x^143+216646363537033464992*x^ 142+744220329731909658512*x^141+120867485146199496098*x^140-\ 3128995586189854735396*x^139-5008700526254897670148*x^138+ 6520728060918258038060*x^137+27003730488873572768634*x^136-\ 9424957986053175553836*x^135-53285920829056960767408*x^134-\ 29361210697794293921980*x^133+61685617178072747431569*x^132+ 114933769106935361329438*x^131+21078456572017392759312*x^130-\ 172233237893531283550646*x^129-232497428427160040247137*x^128+ 25391804036664903839304*x^127+423522604333576368263780*x^126+ 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61245804710302790817904*x^95+18584331618936883572336*x^94-\ 19297441456245722636920*x^93-31659990184899731189388*x^92-\ 19856664566681602257924*x^91+374607479798563501860*x^90+12875257649366789174044 *x^89+11645460436128001015578*x^88+2261385642386307502716*x^87-\ 5132704390598977601200*x^86-4994955192216123203836*x^85-231993761496209637415*x ^84+2389415485918658398580*x^83+597051864167038782936*x^82-\ 1050243526825232582884*x^81+486818645909824635610*x^80+3328020129348522608*x^79 -29180637581781216152*x^78-39748222893216616648*x^77-28532421399289896644*x^76-\ 8889319946404895030*x^75+6351251636928405460*x^74+11307090178522628766*x^73+ 7592355690142818235*x^72+942710027518271316*x^71-3315671417691303276*x^70-\ 3307416087754132556*x^69-762171308203697481*x^68+1269244600920752282*x^67+ 1183353744749412388*x^66-118292950318235602*x^65-695058115708787027*x^64-\ 40694015469133360*x^63+388216849975927252*x^62-175193283136602456*x^61+ 28065574736143647*x^60-1586980732254096*x^59-1601595134903904*x^58-\ 990953015700120*x^57-219362780122152*x^56+298791107599776*x^55+413316359850824* x^54+232974907500216*x^53-13513345356240*x^52-142480592567764*x^51-\ 111653289091008*x^50-5872746203724*x^49+59801224358566*x^48+40434812211928*x^47 -14044374989028*x^46-28494589705192*x^45+3608547892209*x^44+16505745028260*x^43 -10832245214448*x^42+2770545041796*x^41-269406400218*x^40+7922807112*x^39+ 4056755676*x^38-464114888*x^37-2987798012*x^36-2882736754*x^35-1184626700*x^34+ 507507138*x^33+1131221375*x^32+666871312*x^31-130820172*x^30-481337128*x^29-\ 218194513*x^28+182537256*x^27+203381032*x^26-79282792*x^25-126301476*x^24+ 116409916*x^23-41877616*x^22+7416780*x^21-467799*x^20+20444*x^19-9392*x^18-\ 21244*x^17-16138*x^16-3568*x^15+6056*x^14+7592*x^13+2879*x^12-2302*x^11-3440*x^ 10-754*x^9+1807*x^8+1204*x^7-992*x^6-812*x^5+1129*x^4-542*x^3+136*x^2-18*x+1) Theorem number, 19, Let a(n) be the, 20, -bonacci numbers that are defined via the generating function infinity ----- \ n 20 19 18 17 16 15 14 13 12 ) a(n) x = x/(-x - x - x - x - x - x - x - x - x / ----- n = 0 11 10 9 8 7 6 5 4 3 2 - x - x - x - x - x - x - x - x - x - x - x + 1) Let b(n) be the binomial convolution n ----- \ b(n) = ) binomial(n, r) a(r) a(n - r) / ----- r = 0 Then the generating function of the sequence b(n) is infinity ----- \ n 2 199 198 ) b(n) x = 2 x (6710886400000000 x - 45214597120000000 x / ----- n = 0 197 196 - 10938535116800000 x + 540449924710400000 x 195 194 + 438057520470425600 x - 5071441065651650560 x 193 192 - 12123996794401041408 x + 31330945396830325056 x 191 190 + 186072851061268438228 x + 36433452302400305246 x 189 188 - 1910661493222349735836 x - 4324675883868640815236 x 187 186 + 11572467443538510696834 x + 70731883211402347587418 x 185 184 + 23234295806460869297204 x - 710537407793311171926546 x 183 182 - 1661363885938306946715160 x + 5663085151532457979299438 x 181 180 + 57812618422580221034435790 x - 58249995703621018338516622 x 179 178 - 308257280545333903847912058 x - 25196689221903747508431764 x 177 176 + 1056520776818942922030690912 x + 1251819942624092757358027396 x 175 174 - 1835048981486252705301467380 x - 6029101529068119006825125098 x 173 172 - 2037367640430662186070606744 x + 15098172503775814058737133200 x 171 170 + 25084817737878480026764677885 x - 11478433407588877616511811457 x 169 168 - 86677096213203985098060210384 x - 78296449955296541339768583967 x 167 + 148494275520486465179043200826 x 166 165 + 417802810016830923701619791459 x + 99151601262153365003023638029 x 164 - 1148221654865412118023043195069 x 163 - 1790339909662084862621587155645 x 162 + 1646842999648609797284696432886 x 161 + 4663457563298695948505004657916 x 160 + 1537646443419054724856280193380 x 159 - 5762906638164987961570655576819 x 158 - 8382631330251170038030334342312 x 157 - 710711134245601013354876592604 x 156 + 11180215125997102379554187421374 x 155 + 13559233156384918799306585454895 x 154 - 142815007616200367402577490319 x 153 - 18738847829488701132636518910553 x 152 - 21090526659404635625337390095667 x 151 + 1374951842434977394358393393409 x 150 + 29895967792542708232499796053766 x 149 + 31769523787031351913399183052606 x 148 - 4524525120332808766655518140032 x 147 - 47225350768229609592849510221947 x 146 - 44613741083219259832424961695620 x 145 + 15235737180330771378571900311194 x 144 + 68201043152410215872912968809256 x 143 + 35151361630105716913539852100303 x 142 - 24262685444407537674624138324224 x 141 - 51210487954939625091314211088674 x 140 - 31250687602103167037738681905299 x 139 + 9728809369222768414236600168798 x 138 + 35724966307411925547083515135148 x 137 + 29850227840570286715234899935459 x 136 + 2656832640270462444225353908270 x 135 - 21434512910920092652684572232629 x 134 - 25020820032171731394040144405323 x 133 - 9293986041828190571271481958239 x 132 + 10284868446954261209569382159584 x 131 + 18142013406139657842331808090500 x 130 + 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1922995074 x - 723305443 x + 496031730 x 30 29 28 27 + 821478497 x + 256219951 x - 380022961 x - 337335946 x 26 25 24 23 + 169683868 x + 224024236 x - 223403969 x + 82888386 x 22 21 20 19 18 - 14613672 x + 1092711 x + 13092 x - 10222 x - 19614 x 17 16 15 14 13 12 - 16548 x - 6946 x + 2356 x + 6696 x + 5279 x + 832 x 11 10 9 8 7 6 5 - 2625 x - 2681 x - 204 x + 1689 x + 922 x - 970 x - 693 x 4 3 2 / 20 19 + 1066 x - 529 x + 135 x - 18 x + 1) / ((1048576 x + 524288 x / 18 17 16 15 14 13 + 262144 x + 131072 x + 65536 x + 32768 x + 16384 x + 8192 x 12 11 10 9 8 7 6 + 4096 x + 2048 x + 1024 x + 512 x + 256 x + 128 x + 64 x 5 4 3 2 190 189 188 + 32 x + 16 x + 8 x + 4 x + 2 x - 1) (x + 30 x + 620 x 187 186 185 184 + 11865 x + 225295 x + 4280976 x + 81369281 x 183 182 181 + 1546620258 x + 29397056539 x + 558757672887 x 180 179 178 - 9859543416787 x + 38025868524341 x + 23376347416874 x 177 176 175 - 308476986632500 x - 383198006044466 x + 2095365631682988 x 174 173 172 + 6931121593746381 x - 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