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A146765
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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 - 1)*Sum[Binomial[n, m]*x^m*(1 + x^(n - 2*m)), {m, 1, n - 1}]].
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1, 1, 1, 1, 10, 1, 1, 27, 27, 1, 1, 68, 102, 68, 1, 1, 165, 330, 330, 165, 1, 1, 390, 975, 1300, 975, 390, 1, 1, 903, 2709, 4515, 4515, 2709, 903, 1, 1, 2056, 7196, 14392, 17990, 14392, 7196, 2056, 1, 1, 4617, 18468, 43092, 64638, 64638, 43092, 18468, 4617, 1, 1
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Row sums are:{1, 2, 12, 56, 240, 992, 4032, 16256, 65280, 261632, 1047552}.
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FORMULA
| p(x,n)=If[n == 0, 1, (x + 1)^n + 2^(n - 1)*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, 10, 1}, {1, 27, 27, 1}, {1, 68, 102, 68, 1}, {1, 165, 330, 330, 165, 1}, {1, 390, 975, 1300, 975, 390, 1}, {1, 903, 2709, 4515, 4515, 2709, 903, 1}, {1, 2056, 7196, 14392, 17990, 14392, 7196, 2056, 1}, {1, 4617, 18468, 43092, 64638, 64638, 43092, 18468, 4617, 1}, {1, 10250, 46125, 123000, 215250, 258300, 215250, 123000, 46125, 10250, 1}
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MATHEMATICA
| Clear[p, x, n]; p[x_, n_] = If[ n == 0, 1, (x + 1)^n + 2^(n - 1)*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: A113280 A159041 A154979 * A190152 A154984 A173047
Adjacent sequences: A146762 A146763 A146764 * A146766 A146767 A146768
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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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