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A146766
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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 - 2)*Sum[Binomial[n, m]*x^m*(1 + x^(n - 2*m)), {m, 1, n - 1}]].
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0
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1, 1, 1, 1, 6, 1, 1, 15, 15, 1, 1, 36, 54, 36, 1, 1, 85, 170, 170, 85, 1, 1, 198, 495, 660, 495, 198, 1, 1, 455, 1365, 2275, 2275, 1365, 455, 1, 1, 1032, 3612, 7224, 9030, 7224, 3612, 1032, 1, 1, 2313, 9252, 21588, 32382, 32382, 21588, 9252, 2313, 1, 1, 5130, 23085
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Row sums are:{1, 2, 8, 32, 128, 512, 2048, 8192, 32768, 131072, 524288}.
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FORMULA
| p(x,n)=If[n == 0, 1, (x + 1)^n + 2^(n - 2)*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, 6, 1}, {1, 15, 15, 1}, {1, 36, 54, 36, 1}, {1, 85, 170, 170, 85, 1}, {1, 198, 495, 660, 495, 198, 1}, {1, 455, 1365, 2275, 2275, 1365, 455, 1}, {1, 1032, 3612, 7224, 9030, 7224, 3612, 1032, 1}, {1, 2313, 9252, 21588, 32382, 32382, 21588, 9252, 2313, 1}, {1, 5130, 23085, 61560, 107730, 129276, 107730, 61560, 23085, 5130, 1}
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MATHEMATICA
| Clear[p, x, n]; p[x_, n_] = If[ n == 0, 1, (x + 1)^n + 2^(n - 2)*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: A168291 A154980 A166344 * A176152 A146958 A154653
Adjacent sequences: A146763 A146764 A146765 * A146767 A146768 A146769
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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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