0,1

Row sums are:

{2, 6, 36, 324, 3888, 58320, 1049760, 22044960, 529079040, 14285134080,

428554022400,...}.

This results from a modular form bilinear approach summed:

f1(x)=(3*x+1)/(-x); f2(x)=(3*x+2)/(-x).

p(x,n)=(-1)^(-1 + n)* 3^n* (-1 + x)(1 - n) LerchPhi[x, -n, 1/3]+

(-1)^(-1 + n)* 3^n* (-1 + x)(1 - n) LerchPhi[x, -n, 2/3];

p(x,n)=(-1)^(n + 1)*(x - 1)^(n + 1)*Sum[(3*m + 2)^n*x^m, {m, 0, Infinity}] +

(-1)^(n + 1)*(x - 1)^(n + 1)*Sum[(3*m + 1)^n*x^m, {m, 0, Infinity}];

t(n,m)=coefficients(p(x,n))

{2},

{3, 3},

{5, 26, 5},

{9, 153, 153, 9},

{17, 796, 2262, 796, 17},

{33, 3951, 25176, 25176, 3951, 33},

{65, 19266, 243111, 524876, 243111, 19266, 65},

{129, 93477, 2168235, 8760639, 8760639, 2168235, 93477, 129},

{257, 453848, 18445820, 127880936, 235517318, 127880936, 18445820, 453848, 257}, {513, 2210139, 152441730, 1711859886, 5276054772, 5276054772, 1711859886,152441730, 2210139, 513},

{1025, 10802926, 1237317237, 21613648728, 104609410314, 173611661940,104609410314, 21613648728, 1237317237, 10802926, 1025}

Clear[p]; p[x_, n_] = (-1)^(n + 1)*(x - 1)^(n + 1)*Sum[(3*m + 2)^n*x^m, {m, 0, Infinity}]

+ (-1)^(n + 1)*(x - 1)^(n + 1)*Sum[(3*m + 1)^n*x^m, {m, 0, Infinity}];

Table[FullSimplify[ExpandAll[p[x, n]]], {n, 0, 10}];

Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}];

Flatten[%]

Sequence in context: A064776 A096659 A154695 this_sequence A046826 A054892 A104570

Adjacent sequences: A154643 A154644 A154645 this_sequence A154647 A154648 A154649

nonn,uned,tabl

Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 13 2009