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A146769
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Coefficients of polynomial P(n) by rows, with P(n) = (x+1)^n + 2^(n-3)*((x+1)^n - x^n - 1) for n > 0 and P(0) = 1.
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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
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OFFSET
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0,5
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COMMENTS
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Original name: 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}]].
Row sums are:{1, 2, 5, 14, 44, 152, 560, 2144, 8384, 33152, 131840}.
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LINKS
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FORMULA
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G.f.: y/(4*(2*y-1)) - 1/(x*y+y-1) - 1/(8*(2*x*y+2*y-1)) + 1/(8*(2*x*y-1)). - Robert Israel, Apr 29 2015
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EXAMPLE
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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, 27090, 15480, 5805, 1290, 1;
...
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MATHEMATICA
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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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PROG
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(Magma) /* As triangle: */ [1]; for n in [1..10] do; R<x> := PolynomialAlgebra(RationalField(), n); Coefficients((x+1)^n + 2^(n-3)*((x+1)^n - x^n - 1)); end for; // Bruno Berselli, Apr 30 2015
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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