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A146767
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A new symmetrical polynomial form to give a triangle sequence: p(x,n)=If[n == 0, 1, (x + 1)^n + 2^(n - 3)*Sum[Binomial[n, m]*x^m*(1 + x^(n - 2*m)), {m, 1, n - 1}]].
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1, 1, 1, 1, 4, 1, 1, 9, 9, 1, 1, 20, 30, 20, 1, 1, 45, 90, 90, 45, 1, 1, 102, 255, 340, 255, 102, 1, 1, 231, 693, 1155, 1155, 693, 231, 1, 1, 520, 1820, 3640, 4550, 3640, 1820, 520, 1, 1, 1161, 4644, 10836, 16254, 16254, 10836, 4644, 1161, 1, 1, 2570, 11565, 30840, 53970
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Row sums are:{1, 2, 6, 20, 72, 272, 1056, 4160, 16512, 65792, 262656}.
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FORMULA
| p(x,n)=If[n == 0, 1, (x + 1)^n + 2^(n - 3)*Sum[Binomial[n, m]*x^m*(1 + x^(n - 2*m)), {m, 1, n - 1}]]; t(n,m)=coefficients(p(x,n)).
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EXAMPLE
| {1}, {1, 1}, {1, 4, 1}, {1, 9, 9, 1}, {1, 20, 30, 20, 1}, {1, 45, 90, 90, 45, 1}, {1, 102, 255, 340, 255, 102, 1}, {1, 231, 693, 1155, 1155, 693, 231, 1}, {1, 520, 1820, 3640, 4550, 3640, 1820, 520, 1}, {1, 1161, 4644, 10836, 16254, 16254, 10836, 4644, 1161, 1}, {1, 2570, 11565, 30840, 53970, 64764, 53970, 30840, 11565, 2570, 1}
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MATHEMATICA
| Clear[p, x, n]; p[x_, n_] = If[ n == 0, 1, (x + 1)^n + 2^(n - 3)*Sum[Binomial[n, m]*x^m*(1 + x^(n - 2*m)), {m, 1, n - 1}]]; Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}]; Flatten[%]
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CROSSREFS
| Sequence in context: A180960 A157192 A154982 * A146955 A155451 A189280
Adjacent sequences: A146764 A146765 A146766 * A146768 A146769 A146770
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KEYWORD
| nonn
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 02 2008
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