A154649 A triangular sequence of coefficients: p(x,n)=((-1)^(n + 1)*(x - 1)^(n + 1)*Sum[(2*m - 1)^n*x^m, {m, 0, Infinity}] + (-1)^(n + 1)*(x - 1)^(n + 1)*Sum[(2*m + 3)^n*x^m, {m, 0, Infinity}])/2.

1, 1, 1, 5, -2, 5, 13, 11, 11, 13, 41, 108, 86, 108, 41, 121, 837, 962, 962, 837, 121, 365, 5258, 12163, 10508, 12163, 5258, 365, 1093, 30319, 130965, 160183, 160183, 130965, 30319, 1093, 3281, 165784, 1245980, 2503208, 2485414, 2503208, 1245980, 165784

Offset

0,4

Comments

Row sums are:A000165

{1, 2, 8, 48, 384, 3840, 46080, 645120, 10321920, 185794560, 3715891200,...}.

The row sums are equivalent to the MacMahon numbers rows sums.

This results from a modular form bilinear approach summed:

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

Links

Table of n, a(n) for n=0..43.

Formula

p(x,n)=((-1)^(-1 + n)* (-1 + x)(1 - n) *((-1)^n+2*n*LerchPhi[x, -n, 1/2])+

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

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

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

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

Example

{1},
{1, 1},
{5, -2, 5},
{13, 11, 11, 13},
{41, 108, 86, 108, 41},
{121, 837, 962, 962, 837, 121},
{365, 5258, 12163, 10508, 12163, 5258, 365},
{1093, 30319, 130965, 160183, 160183, 130965, 30319, 1093},
{3281, 165784, 1245980, 2503208, 2485414, 2503208, 1245980, 165784, 3281},
{9841, 878153, 10863860, 35584772, 45560654, 45560654, 35584772, 10863860, 878153, 9841},
{29525, 4558038, 89180081, 458019464, 852697082, 906922820, 852697082, 458019464, 89180081, 4558038, 29525}

Mathematica

Clear[p]; p[x_, n_] = ((-1)^(n + 1)*(x - 1)^(n + 1)*Sum[(2*m - 1)^n*x^m, {m, 0, Infinity}]
+ (-1)^(n + 1)*(x - 1)^(n + 1)*Sum[(2*m + 3)^n*x^m, {m, 0, Infinity}])/2;
Table[FullSimplify[ExpandAll[p[x, n]]], {n, 0, 10}];
Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}];
Flatten[%]

Crossrefs

A000165

Sequence in context: A085997 A071546 A378873 * A100040 A197271 A248260

Adjacent sequences: A154646 A154647 A154648 * A154650 A154651 A154652

Keyword

sign,tabl,uned

Author

Roger L. Bagula, Jan 13 2009

Status

approved