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A165891
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Polynomial infinite sum product coefficient triangle: p(x,n)=(1 + x)^n*(1 - x)^(n + 2)*Sum[k^(n + 1)*x^k, {k, 0, Infinity}]/x
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0
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1, 1, 2, 1, 1, 6, 10, 6, 1, 1, 14, 47, 68, 47, 14, 1, 1, 30, 176, 450, 606, 450, 176, 30, 1, 1, 62, 597, 2392, 5162, 6612, 5162, 2392, 597, 62, 1, 1, 126, 1926, 11382, 35967, 69132, 85492, 69132, 35967, 11382, 1926, 126, 1, 1, 254, 6043, 50892, 223785, 600546
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Sequence based on the Euler number A008292 infinite sum/PolyLog[ -1-n,x]/x .
Row sums are:
{1, 4, 24, 192, 1920, 23040, 322560, 5160960, 92897280, 1857945600,
40874803200}
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FORMULA
| Alternative form:
p(x,n)=(1 + x)^n*(1 - x)^(n + 2)*PolyLog[ -1-n,x]/x
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EXAMPLE
| {1},
{1, 2, 1},
{1, 6, 10, 6, 1},
{1, 14, 47, 68, 47, 14, 1},
{1, 30, 176, 450, 606, 450, 176, 30, 1},
{1, 62, 597, 2392, 5162, 6612, 5162, 2392, 597, 62, 1},
{1, 126, 1926, 11382, 35967, 69132, 85492, 69132, 35967, 11382, 1926, 126, 1},
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MATHEMATICA
| Clear[p, x, n, m]
p[x_, n_] = (1 + x)^n*(1 - x)^(n + 2)*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
Sequence in context: A196073 A144089 A172107 * A039763 A094262 A123554
Adjacent sequences: A165888 A165889 A165890 * A165892 A165893 A165894
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KEYWORD
| nonn,uned
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Sep 29 2009
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