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A165883
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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
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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; internal format)
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OFFSET
| 0,3
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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,...}
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FORMULA
| Alternative form is:
p(x,n)=2^n*(1-x)^(2*n+3)*LerchPhi[x,-n,1/2]*PolyLog[ -1-n,x]/x
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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}
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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}]
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CROSSREFS
| A008292, A060187
Sequence in context: A153731 A154989 A064307 * A110905 A205447 A158202
Adjacent sequences: A165880 A165881 A165882 * A165884 A165885 A165886
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KEYWORD
| nonn,uned,tabf
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Sep 29 2009
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