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A146774
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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, m]*x^m*(1 + x^(n - 2*m)), {m, 1, n - 1}]].
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1, 1, 1, 1, 18, 1, 1, 35, 35, 1, 1, 100, 70, 100, 1, 1, 261, 202, 202, 261, 1, 1, 646, 783, 276, 783, 646, 1, 1, 1543, 2581, 1059, 1059, 2581, 1543, 1, 1, 3592, 7708, 5176, 1094, 5176, 7708, 3592, 1, 1, 8201, 21540, 20564, 5246, 5246, 20564, 21540, 8201, 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, 20, 72, 272, 928, 3136, 10368, 34048, 111104, 361472}.
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FORMULA
| p(x,n)=If[n == 0, 1, (x + 1)^n + 2^(n+1)*Sum[Binomial[n-m, 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, 18, 1}, {1, 35, 35, 1}, {1, 100, 70, 100, 1}, {1, 261, 202, 202, 261, 1}, {1, 646, 783, 276, 783, 646, 1}, {1, 1543, 2581, 1059, 1059, 2581, 1543, 1}, {1, 3592, 7708, 5176, 1094, 5176, 7708, 3592, 1}, {1, 8201, 21540, 20564, 5246, 5246, 20564, 21540, 8201, 1}, {1, 18442, 57389, 71800, 30930, 4348, 30930, 71800, 57389, 18442, 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, 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: A040325 A040324 A168623 * A174451 A144405 A202671
Adjacent sequences: A146771 A146772 A146773 * A146775 A146776 A146777
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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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