This is a table of J[r]^(k)(1,x) for SYMBOLIC k, defined in Richard Stanley\ 's paper arXiv:2101.02131v2 Let F^(k)_i be the k-bonacci numbers and let Prod( 1+t*x^F^(k)_{i+k-1), i=1..n)= Sum(a(n,k)*x^k,k=0..infinity) and let v_r(n,t) be the sum of the r-th power of the coefficients Let J[r,k] be the (ordinary) generating function with respect to x (no relat\ ion to the previous x) J[2, k](1, x), equals k 1 - 2 x ---------------------------- k (k + 1) -2 x + 1 - 2 x + 2 x and in Maple format (1-2*x^k)/(-2*x+1-2*x^k+2*x^(k+1)) J[3, k](1, x), equals k 1 - 4 x ---------------------------- k (k + 1) -2 x + 1 - 4 x + 2 x and in Maple format (1-4*x^k)/(-2*x+1-4*x^k+2*x^(k+1)) J[4, k](1, x), equals k (2 k) 1 - 7 x - 2 x ----------------------------------------- k (2 k) (2 k + 1) -2 x + 1 - 7 x - 2 x + 2 x and in Maple format (1-7*x^k-2*x^(2*k))/(-2*x+1-7*x^k-2*x^(2*k)+2*x^(2*k+1)) J[5, k](1, x), equals k (2 k) 1 - 11 x - 20 x --------------------------------------------------------- k (k + 1) (2 k) (2 k + 1) -2 x + 1 - 11 x - 8 x - 20 x + 10 x and in Maple format (1-11*x^k-20*x^(2*k))/(-2*x+1-11*x^k-8*x^(k+1)-20*x^(2*k)+10*x^(2*k+1)) J[6, k](1, x), equals k (2 k) (3 k) / k (k + 1) (1 - 17 x - 88 x - 4 x ) / (-2 x + 1 - 17 x - 28 x / (2 k) (2 k + 1) (3 k) (3 k + 1) - 88 x + 26 x - 4 x + 4 x ) and in Maple format (1-17*x^k-88*x^(2*k)-4*x^(3*k))/(-2*x+1-17*x^k-28*x^(k+1)-88*x^(2*k)+26*x^(2*k+ 1)-4*x^(3*k)+4*x^(3*k+1)) J[7, k](1, x), equals k (2 k) (3 k) / k (k + 1) (1 - 26 x - 311 x - 84 x ) / (-2 x + 1 - 26 x - 74 x / (2 k) (2 k + 1) (3 k) (3 k + 1) - 311 x + 34 x - 84 x + 42 x ) and in Maple format (1-26*x^k-311*x^(2*k)-84*x^(3*k))/(-2*x+1-26*x^k-74*x^(k+1)-311*x^(2*k)+34*x^(2 *k+1)-84*x^(3*k)+42*x^(3*k+1)) J[8, k](1, x), equals k (2 k) (3 k) (4 k) / k (1 - 40 x - 969 x - 428 x - 4 x ) / (1 - 2 x - 40 x / (k + 1) (2 k) (2 k + 1) (3 k) (3 k + 1) - 174 x - 969 x - 2 x - 428 x + 174 x (4 k) (4 k + 1) - 4 x + 4 x ) and in Maple format (1-40*x^k-969*x^(2*k)-428*x^(3*k)-4*x^(4*k))/(1-2*x-40*x^k-174*x^(k+1)-969*x^(2 *k)-2*x^(2*k+1)-428*x^(3*k)+174*x^(3*k+1)-4*x^(4*k)+4*x^(4*k+1)) J[9, k](1, x), equals k (2 k) (3 k) / k (k + 1) (1 - 62 x - 2819 x - 900 x ) / (1 - 2 x - 62 x - 386 x / (2 k) (2 k + 1) (3 k) (3 k + 1) - 2819 x - 62 x - 900 x + 450 x ) and in Maple format (1-62*x^k-2819*x^(2*k)-900*x^(3*k))/(1-2*x-62*x^k-386*x^(k+1)-2819*x^(2*k)-62*x ^(2*k+1)-900*x^(3*k)+450*x^(3*k+1)) J[10, k](1, x), equals k (2 k) (3 k) (4 k) / k (1 - 96 x - 7945 x - 1852 x - 4 x ) / (-2 x + 1 - 96 x / (k + 1) (2 k) (2 k + 1) (3 k) (3 k + 1) - 830 x - 7945 x - 2 x - 1852 x + 830 x (4 k) (4 k + 1) - 4 x + 4 x ) and in Maple format (1-96*x^k-7945*x^(2*k)-1852*x^(3*k)-4*x^(4*k))/(-2*x+1-96*x^k-830*x^(k+1)-7945* x^(2*k)-2*x^(2*k+1)-1852*x^(3*k)+830*x^(3*k+1)-4*x^(4*k)+4*x^(4*k+1)) J[11, k](1, x), equals k (2 k) (3 k) (4 k) / (-1 + 153 x + 21249 x - 86213 x - 18348 x ) / (2 x - 1 / k (k + 1) (2 k) (2 k + 1) (3 k) + 153 x + 1740 x + 21249 x - 9432 x - 86213 x (3 k + 1) (4 k) (4 k + 1) - 1484 x - 18348 x + 9174 x ) and in Maple format (-1+153*x^k+21249*x^(2*k)-86213*x^(3*k)-18348*x^(4*k))/(2*x-1+153*x^k+1740*x^(k +1)+21249*x^(2*k)-9432*x^(2*k+1)-86213*x^(3*k)-1484*x^(3*k+1)-18348*x^(4*k)+ 9174*x^(4*k+1)) J[12, k](1, x), equals k (2 k) (3 k) (4 k) (5 k) (1 - 243 x - 56447 x + 667319 x + 9582 x - 15404 x (6 k) / k (k + 1) (2 k) - 8 x ) / (-2 x + 1 - 243 x - 3608 x - 56447 x / (2 k + 1) (3 k) (3 k + 1) (4 k) + 61236 x + 667319 x - 3608 x + 9582 x (4 k + 1) (5 k) (5 k + 1) (6 k) - 61242 x - 15404 x + 7216 x - 8 x (6 k + 1) + 8 x ) and in Maple format (1-243*x^k-56447*x^(2*k)+667319*x^(3*k)+9582*x^(4*k)-15404*x^(5*k)-8*x^(6*k))/( -2*x+1-243*x^k-3608*x^(k+1)-56447*x^(2*k)+61236*x^(2*k+1)+667319*x^(3*k)-3608*x ^(3*k+1)+9582*x^(4*k)-61242*x^(4*k+1)-15404*x^(5*k)+7216*x^(5*k+1)-8*x^(6*k)+8* x^(6*k+1)) J[13, k](1, x), equals k (2 k) (3 k) (4 k) (5 k) (1 - 388 x - 148038 x + 4165856 x - 1565891 x - 289380 x ) / k (k + 1) (2 k) (2 k + 1) / (-2 x + 1 - 388 x - 7414 x - 148038 x + 317916 x / (3 k) (3 k + 1) (4 k) (4 k + 1) + 4165856 x - 136252 x - 1565891 x - 318938 x (5 k) (5 k + 1) - 289380 x + 144690 x ) and in Maple format (1-388*x^k-148038*x^(2*k)+4165856*x^(3*k)-1565891*x^(4*k)-289380*x^(5*k))/(-2*x +1-388*x^k-7414*x^(k+1)-148038*x^(2*k)+317916*x^(2*k+1)+4165856*x^(3*k)-136252* x^(3*k+1)-1565891*x^(4*k)-318938*x^(4*k+1)-289380*x^(5*k)+144690*x^(5*k+1)) J[14, k](1, x), equals k (2 k) (3 k) (4 k) (5 k) (1 - 621 x - 385463 x + 22761161 x + 2116566 x - 63044 x (6 k) / k (k + 1) (2 k) - 8 x ) / (-2 x + 1 - 621 x - 15140 x - 385463 x / (2 k + 1) (3 k) (3 k + 1) (4 k) + 1443744 x + 22761161 x - 15140 x + 2116566 x (4 k + 1) (5 k) (5 k + 1) (6 k) - 1443750 x - 63044 x + 30280 x - 8 x (6 k + 1) + 8 x ) and in Maple format (1-621*x^k-385463*x^(2*k)+22761161*x^(3*k)+2116566*x^(4*k)-63044*x^(5*k)-8*x^(6 *k))/(-2*x+1-621*x^k-15140*x^(k+1)-385463*x^(2*k)+1443744*x^(2*k+1)+22761161*x^ (3*k)-15140*x^(3*k+1)+2116566*x^(4*k)-1443750*x^(4*k+1)-63044*x^(5*k)+30280*x^( 5*k+1)-8*x^(6*k)+8*x^(6*k+1)) J[15, k](1, x), equals k (2 k) (3 k) (4 k) (1 - 1000 x - 994458 x + 119408756 x - 134208623 x (5 k) / k (k + 1) - 16637076 x ) / (-2 x + 1 - 1000 x - 30766 x / (2 k) (2 k + 1) (3 k) - 994458 x + 6188172 x + 119408756 x (3 k + 1) (4 k) (4 k + 1) - 8289820 x - 134208623 x - 6186122 x (5 k) (5 k + 1) - 16637076 x + 8318538 x ) and in Maple format (1-1000*x^k-994458*x^(2*k)+119408756*x^(3*k)-134208623*x^(4*k)-16637076*x^(5*k) )/(-2*x+1-1000*x^k-30766*x^(k+1)-994458*x^(2*k)+6188172*x^(2*k+1)+119408756*x^( 3*k)-8289820*x^(3*k+1)-134208623*x^(4*k)-6186122*x^(4*k+1)-16637076*x^(5*k)+ 8318538*x^(5*k+1)) J[16, k](1, x), equals k (2 k) (3 k) (4 k) (1 - 1611 x - 2559407 x + 585266591 x + 44606766 x (5 k) (6 k) / k (k + 1) - 255692 x - 8 x ) / (-2 x + 1 - 1611 x - 62312 x / (2 k) (2 k + 1) (3 k) - 2559407 x + 24862788 x + 585266591 x (3 k + 1) (4 k) (4 k + 1) (5 k) - 62312 x + 44606766 x - 24862794 x - 255692 x (5 k + 1) (6 k) (6 k + 1) + 124624 x - 8 x + 8 x ) and in Maple format (1-1611*x^k-2559407*x^(2*k)+585266591*x^(3*k)+44606766*x^(4*k)-255692*x^(5*k)-8 *x^(6*k))/(-2*x+1-1611*x^k-62312*x^(k+1)-2559407*x^(2*k)+24862788*x^(2*k+1)+ 585266591*x^(3*k)-62312*x^(3*k+1)+44606766*x^(4*k)-24862794*x^(4*k+1)-255692*x^ (5*k)+124624*x^(5*k+1)-8*x^(6*k)+8*x^(6*k+1)) J[17, k](1, x), equals k (2 k) (3 k) (4 k) (1 - 2599 x - 6569850 x + 2764163954 x + 15022392733 x (5 k) (6 k) / k - 15329386299 x - 1347896340 x ) / (-2 x + 1 - 2599 x / (k + 1) (2 k) (2 k + 1) - 125872 x - 6569850 x + 96034590 x (3 k) (3 k + 1) (4 k) + 2764163954 x + 643026032 x + 15022392733 x (4 k + 1) (5 k) (5 k + 1) - 769974566 x - 15329386299 x - 642908352 x (6 k) (6 k + 1) - 1347896340 x + 673948170 x ) and in Maple format (1-2599*x^k-6569850*x^(2*k)+2764163954*x^(3*k)+15022392733*x^(4*k)-15329386299* x^(5*k)-1347896340*x^(6*k))/(-2*x+1-2599*x^k-125872*x^(k+1)-6569850*x^(2*k)+ 96034590*x^(2*k+1)+2764163954*x^(3*k)+643026032*x^(3*k+1)+15022392733*x^(4*k)-\ 769974566*x^(4*k+1)-15329386299*x^(5*k)-642908352*x^(5*k+1)-1347896340*x^(6*k)+ 673948170*x^(6*k+1)) J[18, k](1, x), equals k (2 k) (3 k) (4 k) (1 - 4196 x - 16841706 x + 12680716224 x + 275807280985 x (5 k) (6 k) (7 k) (8 k) + 45547388948 x + 1371988544 x - 2063568 x - 16 x ) / k (k + 1) (2 k) / (-2 x + 1 - 4196 x - 253750 x - 16841706 x / (2 k + 1) (3 k) (3 k + 1) + 359838840 x + 12680716224 x + 10092724500 x (4 k) (4 k + 1) (5 k) + 275807280985 x + 359838838 x + 45547388948 x (5 k + 1) (6 k) (6 k + 1) - 10093485750 x + 1371988544 x - 719677692 x (7 k) (7 k + 1) (8 k) (8 k + 1) - 2063568 x + 1015000 x - 16 x + 16 x ) and in Maple format (1-4196*x^k-16841706*x^(2*k)+12680716224*x^(3*k)+275807280985*x^(4*k)+ 45547388948*x^(5*k)+1371988544*x^(6*k)-2063568*x^(7*k)-16*x^(8*k))/(-2*x+1-4196 *x^k-253750*x^(k+1)-16841706*x^(2*k)+359838840*x^(2*k+1)+12680716224*x^(3*k)+ 10092724500*x^(3*k+1)+275807280985*x^(4*k)+359838838*x^(4*k+1)+45547388948*x^(5 *k)-10093485750*x^(5*k+1)+1371988544*x^(6*k)-719677692*x^(6*k+1)-2063568*x^(7*k )+1015000*x^(7*k+1)-16*x^(8*k)+16*x^(8*k+1)) J[19, k](1, x), equals k (2 k) (3 k) (4 k) (1 - 6782 x - 43115177 x + 57102085832 x + 3287973427007 x (5 k) (6 k) (7 k) / - 1820299893334 x - 1490826289911 x - 75808868436 x ) / ( / k (k + 1) (2 k) -2 x + 1 - 6782 x - 510722 x - 43115177 x (2 k + 1) (3 k) (3 k + 1) + 1319512222 x + 57102085832 x + 103492425230 x (4 k) (4 k + 1) (5 k) + 3287973427007 x - 70431413590 x - 1820299893334 x (5 k + 1) (6 k) (6 k + 1) - 141396315958 x - 1490826289911 x + 69111868602 x (7 k) (7 k + 1) - 75808868436 x + 37904434218 x ) and in Maple format (1-6782*x^k-43115177*x^(2*k)+57102085832*x^(3*k)+3287973427007*x^(4*k)-\ 1820299893334*x^(5*k)-1490826289911*x^(6*k)-75808868436*x^(7*k))/(-2*x+1-6782*x ^k-510722*x^(k+1)-43115177*x^(2*k)+1319512222*x^(2*k+1)+57102085832*x^(3*k)+ 103492425230*x^(3*k+1)+3287973427007*x^(4*k)-70431413590*x^(4*k+1)-\ 1820299893334*x^(5*k)-141396315958*x^(5*k+1)-1490826289911*x^(6*k)+69111868602* x^(6*k+1)-75808868436*x^(7*k)+37904434218*x^(7*k+1)) J[20, k](1, x), equals k (2 k) (3 k) (4 k) (1 - 10964 x - 110369322 x + 252584574960 x + 34979269477849 x (5 k) (6 k) (7 k) (8 k) + 2366125043732 x + 18550240640 x - 8300880 x - 16 x ) / k (k + 1) (2 k) / (-2 x + 1 - 10964 x - 1026646 x - 110369322 x / (2 k + 1) (3 k) (3 k + 1) + 4747929480 x + 252584574960 x + 930476920260 x (4 k) (4 k + 1) (5 k) + 34979269477849 x + 4747929478 x + 2366125043732 x (5 k + 1) (6 k) (6 k + 1) - 930480000198 x + 18550240640 x - 9495858972 x (7 k) (7 k + 1) (8 k) (8 k + 1) - 8300880 x + 4106584 x - 16 x + 16 x ) and in Maple format (1-10964*x^k-110369322*x^(2*k)+252584574960*x^(3*k)+34979269477849*x^(4*k)+ 2366125043732*x^(5*k)+18550240640*x^(6*k)-8300880*x^(7*k)-16*x^(8*k))/(-2*x+1-\ 10964*x^k-1026646*x^(k+1)-110369322*x^(2*k)+4747929480*x^(2*k+1)+252584574960*x ^(3*k)+930476920260*x^(3*k+1)+34979269477849*x^(4*k)+4747929478*x^(4*k+1)+ 2366125043732*x^(5*k)-930480000198*x^(5*k+1)+18550240640*x^(6*k)-9495858972*x^( 6*k+1)-8300880*x^(7*k)+4106584*x^(7*k+1)-16*x^(8*k)+16*x^(8*k+1)) ---------------------------- To sum-up, the first, 19, terms starting at r=2 are [(1-2*x^k)/(-2*x+1-2*x^k+2*x^(k+1)), (1-4*x^k)/(-2*x+1-4*x^k+2*x^(k+1)), (1-7*x ^k-2*x^(2*k))/(-2*x+1-7*x^k-2*x^(2*k)+2*x^(2*k+1)), (1-11*x^k-20*x^(2*k))/(-2*x +1-11*x^k-8*x^(k+1)-20*x^(2*k)+10*x^(2*k+1)), (1-17*x^k-88*x^(2*k)-4*x^(3*k))/( -2*x+1-17*x^k-28*x^(k+1)-88*x^(2*k)+26*x^(2*k+1)-4*x^(3*k)+4*x^(3*k+1)), (1-26* x^k-311*x^(2*k)-84*x^(3*k))/(-2*x+1-26*x^k-74*x^(k+1)-311*x^(2*k)+34*x^(2*k+1)-\ 84*x^(3*k)+42*x^(3*k+1)), (1-40*x^k-969*x^(2*k)-428*x^(3*k)-4*x^(4*k))/(1-2*x-\ 40*x^k-174*x^(k+1)-969*x^(2*k)-2*x^(2*k+1)-428*x^(3*k)+174*x^(3*k+1)-4*x^(4*k)+ 4*x^(4*k+1)), (1-62*x^k-2819*x^(2*k)-900*x^(3*k))/(1-2*x-62*x^k-386*x^(k+1)-\ 2819*x^(2*k)-62*x^(2*k+1)-900*x^(3*k)+450*x^(3*k+1)), (1-96*x^k-7945*x^(2*k)-\ 1852*x^(3*k)-4*x^(4*k))/(-2*x+1-96*x^k-830*x^(k+1)-7945*x^(2*k)-2*x^(2*k+1)-\ 1852*x^(3*k)+830*x^(3*k+1)-4*x^(4*k)+4*x^(4*k+1)), (-1+153*x^k+21249*x^(2*k)-\ 86213*x^(3*k)-18348*x^(4*k))/(2*x-1+153*x^k+1740*x^(k+1)+21249*x^(2*k)-9432*x^( 2*k+1)-86213*x^(3*k)-1484*x^(3*k+1)-18348*x^(4*k)+9174*x^(4*k+1)), (1-243*x^k-\ 56447*x^(2*k)+667319*x^(3*k)+9582*x^(4*k)-15404*x^(5*k)-8*x^(6*k))/(-2*x+1-243* x^k-3608*x^(k+1)-56447*x^(2*k)+61236*x^(2*k+1)+667319*x^(3*k)-3608*x^(3*k+1)+ 9582*x^(4*k)-61242*x^(4*k+1)-15404*x^(5*k)+7216*x^(5*k+1)-8*x^(6*k)+8*x^(6*k+1) ), (1-388*x^k-148038*x^(2*k)+4165856*x^(3*k)-1565891*x^(4*k)-289380*x^(5*k))/(-\ 2*x+1-388*x^k-7414*x^(k+1)-148038*x^(2*k)+317916*x^(2*k+1)+4165856*x^(3*k)-\ 136252*x^(3*k+1)-1565891*x^(4*k)-318938*x^(4*k+1)-289380*x^(5*k)+144690*x^(5*k+ 1)), (1-621*x^k-385463*x^(2*k)+22761161*x^(3*k)+2116566*x^(4*k)-63044*x^(5*k)-8 *x^(6*k))/(-2*x+1-621*x^k-15140*x^(k+1)-385463*x^(2*k)+1443744*x^(2*k+1)+ 22761161*x^(3*k)-15140*x^(3*k+1)+2116566*x^(4*k)-1443750*x^(4*k+1)-63044*x^(5*k )+30280*x^(5*k+1)-8*x^(6*k)+8*x^(6*k+1)), (1-1000*x^k-994458*x^(2*k)+119408756* x^(3*k)-134208623*x^(4*k)-16637076*x^(5*k))/(-2*x+1-1000*x^k-30766*x^(k+1)-\ 994458*x^(2*k)+6188172*x^(2*k+1)+119408756*x^(3*k)-8289820*x^(3*k+1)-134208623* x^(4*k)-6186122*x^(4*k+1)-16637076*x^(5*k)+8318538*x^(5*k+1)), (1-1611*x^k-\ 2559407*x^(2*k)+585266591*x^(3*k)+44606766*x^(4*k)-255692*x^(5*k)-8*x^(6*k))/(-\ 2*x+1-1611*x^k-62312*x^(k+1)-2559407*x^(2*k)+24862788*x^(2*k+1)+585266591*x^(3* k)-62312*x^(3*k+1)+44606766*x^(4*k)-24862794*x^(4*k+1)-255692*x^(5*k)+124624*x^ (5*k+1)-8*x^(6*k)+8*x^(6*k+1)), (1-2599*x^k-6569850*x^(2*k)+2764163954*x^(3*k)+ 15022392733*x^(4*k)-15329386299*x^(5*k)-1347896340*x^(6*k))/(-2*x+1-2599*x^k-\ 125872*x^(k+1)-6569850*x^(2*k)+96034590*x^(2*k+1)+2764163954*x^(3*k)+643026032* x^(3*k+1)+15022392733*x^(4*k)-769974566*x^(4*k+1)-15329386299*x^(5*k)-642908352 *x^(5*k+1)-1347896340*x^(6*k)+673948170*x^(6*k+1)), (1-4196*x^k-16841706*x^(2*k )+12680716224*x^(3*k)+275807280985*x^(4*k)+45547388948*x^(5*k)+1371988544*x^(6* k)-2063568*x^(7*k)-16*x^(8*k))/(-2*x+1-4196*x^k-253750*x^(k+1)-16841706*x^(2*k) +359838840*x^(2*k+1)+12680716224*x^(3*k)+10092724500*x^(3*k+1)+275807280985*x^( 4*k)+359838838*x^(4*k+1)+45547388948*x^(5*k)-10093485750*x^(5*k+1)+1371988544*x ^(6*k)-719677692*x^(6*k+1)-2063568*x^(7*k)+1015000*x^(7*k+1)-16*x^(8*k)+16*x^(8 *k+1)), (1-6782*x^k-43115177*x^(2*k)+57102085832*x^(3*k)+3287973427007*x^(4*k)-\ 1820299893334*x^(5*k)-1490826289911*x^(6*k)-75808868436*x^(7*k))/(-2*x+1-6782*x ^k-510722*x^(k+1)-43115177*x^(2*k)+1319512222*x^(2*k+1)+57102085832*x^(3*k)+ 103492425230*x^(3*k+1)+3287973427007*x^(4*k)-70431413590*x^(4*k+1)-\ 1820299893334*x^(5*k)-141396315958*x^(5*k+1)-1490826289911*x^(6*k)+69111868602* x^(6*k+1)-75808868436*x^(7*k)+37904434218*x^(7*k+1)), (1-10964*x^k-110369322*x^ (2*k)+252584574960*x^(3*k)+34979269477849*x^(4*k)+2366125043732*x^(5*k)+ 18550240640*x^(6*k)-8300880*x^(7*k)-16*x^(8*k))/(-2*x+1-10964*x^k-1026646*x^(k+ 1)-110369322*x^(2*k)+4747929480*x^(2*k+1)+252584574960*x^(3*k)+930476920260*x^( 3*k+1)+34979269477849*x^(4*k)+4747929478*x^(4*k+1)+2366125043732*x^(5*k)-\ 930480000198*x^(5*k+1)+18550240640*x^(6*k)-9495858972*x^(6*k+1)-8300880*x^(7*k) +4106584*x^(7*k+1)-16*x^(8*k)+16*x^(8*k+1))] ---------------------------- This took, 68888.093, seconds.