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A146769
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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 - 4)*Sum[Binomial[n, m]*x^m*(1 + x^(n - 2*m)), {m, 1, n - 1}]].
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1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 12, 18, 12, 1, 1, 25, 50, 50, 25, 1, 1, 54, 135, 180, 135, 54, 1, 1, 119, 357, 595, 595, 357, 119, 1, 1, 264, 924, 1848, 2310, 1848, 924, 264, 1, 1, 585, 2340, 5460, 8190, 8190, 5460, 2340, 585, 1, 1, 1290, 5805, 15480, 27090, 32508
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Row sums are:{1, 2, 5, 14, 44, 152, 560, 2144, 8384, 33152, 131840}.
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FORMULA
| p(x,n)=If[n == 0, 1, (x + 1)^n + 2^(n - 4)*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, 3, 1}, {1, 6, 6, 1}, {1, 12, 18, 12, 1}, {1, 25, 50, 50, 25, 1}, {1, 54, 135, 180, 135, 54, 1}, {1, 119, 357, 595, 595, 357, 119, 1}, {1, 264, 924, 1848, 2310, 1848, 924, 264, 1}, {1, 585, 2340, 5460, 8190, 8190, 5460, 2340, 585, 1}, {1, 1290, 5805, 15480, 27090, 32508, 27090, 15480, 5805, 1290, 1}
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MATHEMATICA
| Clear[p, x, n]; p[x_, n_] = If[ n == 0, 1, (x + 1)^n + 2^(n - 4)*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: A131235 A202812 A157243 * A189610 A172427 A143362
Adjacent sequences: A146766 A146767 A146768 * A146770 A146771 A146772
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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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