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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. 0
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 (list; table; graph; refs; listen; history; text; internal format)
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: A201530 A085997 A071546 * A100040 A197271 A248260

Adjacent sequences:  A154646 A154647 A154648 * A154650 A154651 A154652

KEYWORD

sign,tabl,uned

AUTHOR

Roger L. Bagula, Jan 13 2009

STATUS

approved

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Last modified December 4 16:52 EST 2016. Contains 278750 sequences.