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A165883 Polynomial coefficient sequence from the infinite sum: p(x,n)=(1 - x)^(2*n + 3)*Sum[(2*k + 1)^n*x^ k, {k, 0, Infinity}]*Sum[k^(n + 1)*x^k, {k, 0, Infinity}]/x 0
1, 1, 2, 1, 1, 10, 26, 10, 1, 1, 34, 287, 508, 287, 34, 1, 1, 102, 2272, 11098, 19134, 11098, 2272, 102, 1, 1, 294, 15493, 169432, 675706, 1042948, 675706, 169432, 15493, 294, 1, 1, 842, 98374, 2151026, 17138559, 55643460, 82178676, 55643460, 17138559 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The sequence is a product of the Eulerian numbers A008292 and the MacMahon numbers A060187.

The row sums are:

{1, 4, 48, 1152, 46080, 2764800, 232243200, 26011238400, 3745618329600,

674211299328000, 148326485852160000,...}

LINKS

Table of n, a(n) for n=0..44.

FORMULA

Alternative form is:

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

EXAMPLE

{1},

{1, 2, 1},

{1, 10, 26, 10, 1},

{1, 34, 287, 508, 287, 34, 1},

{1, 102, 2272, 11098, 19134, 11098, 2272, 102, 1},

{1, 294, 15493, 169432, 675706, 1042948, 675706, 169432, 15493, 294, 1}

MATHEMATICA

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

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

Flatten[%]

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

CROSSREFS

A008292, A060187

Sequence in context: A262226 A154989 A064307 * A260950 A110905 A205447

Adjacent sequences:  A165880 A165881 A165882 * A165884 A165885 A165886

KEYWORD

nonn,uned,tabf

AUTHOR

Roger L. Bagula, Sep 29 2009

STATUS

approved

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Last modified December 11 21:15 EST 2017. Contains 295919 sequences.