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A165889
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Polynomial infinite sum product coefficient triangle: p(x,n)=(1 - x)^(2*n + 4)*Sum[(k)^(n + 1)*x^ k, {k, 0, Infinity}]*Sum[k^(n + 1)*x^k, {k, 0, Infinity}]/x^2
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0
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1, 1, 2, 1, 1, 8, 18, 8, 1, 1, 22, 143, 244, 143, 22, 1, 1, 52, 808, 3484, 5710, 3484, 808, 52, 1, 1, 114, 3853, 35032, 125746, 188908, 125746, 35032, 3853, 114, 1, 1, 240, 16782, 290672, 2000703, 6040992, 8702820, 6040992, 2000703, 290672, 16782, 240
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Based on the Eulerian number A008292 PolyLog[] infinite sum, this is the square.
Row sums are:
{1, 4, 36, 576, 14400, 518400, 25401600, 1625702400, 131681894400,
13168189440000, 1593350922240000,...}
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FORMULA
| Alternative form:
p(x,n)=(1-x)^(2*n+4)*PolyLog[ -1-n,x]^2/x^2
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EXAMPLE
| {1},
{1, 2, 1},
{1, 8, 18, 8, 1},
{1, 22, 143, 244, 143, 22, 1},
{1, 52, 808, 3484, 5710, 3484, 808, 52, 1},
{1, 114, 3853, 35032, 125746, 188908, 125746, 35032, 3853, 114, 1},
{1, 240, 16782, 290672, 2000703, 6040992, 8702820, 6040992, 2000703, 290672, 16782, 240, 1}
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MATHEMATICA
| Clear[p, x, n, m]
p[x_, n_] = (1 - x)^(2*n + 4)* Sum[(k)^(n + 1)*x^k, {k, 0, Infinity}]*Sum[k^(n + 1)*x^ k, {k, 0, Infinity}]/x^2;
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
| Cf. A008292
Sequence in context: A129276 A156901 A167400 * A087127 A144946 A157109
Adjacent sequences: A165886 A165887 A165888 * A165890 A165891 A165892
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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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