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Search: id:A152300
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| A152300 |
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A symmetrical triangle of coefficients of polynomials: q(x,n)=((1 - x)^(2*n)/(n*x))*Sum[Binomial[k + n - 1, k]*k^n*x^k, {k, 0, Infinity}]; p(x,n)=q(x,n)+x^(n-1)*q(1/x,n); t(n,m)=coefficients(p(x,n)). |
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1
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| 2, 3, 3, 10, 20, 10, 65, 145, 145, 65, 626, 1612, 1572, 1612, 626, 7777, 24549, 23114, 23114, 24549, 7777, 117650, 450564, 496974, 340664, 496974, 450564, 117650, 2097153, 9493425, 12990807, 7851015, 7851015, 12990807, 9493425, 2097153
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Row sums are: {2, 6, 40, 420, 6048, 110880, 2471040, 64864800, 1960358400, 67044257280,...}
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FORMULA
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q(x,n)=((1 - x)^(2*n)/(n*x))*Sum[Binomial[k + n - 1, k]*k^n*x^k, {k, 0, Infinity}];
p(x,n)=q(x,n)+x^(n-1)*q(1/x,n);
t(n,m)=coefficients(p(x,n)).
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EXAMPLE
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{2},
{3, 3},
{10, 20, 10},
{65, 145, 145, 65},
{626, 1612, 1572, 1612, 626},
{7777, 24549, 23114, 23114, 24549, 7777},
{117650, 450564, 496974, 340664, 496974, 450564, 117650},
{2097153, 9493425, 12990807, 7851015, 7851015, 12990807, 9493425, 2097153},
{43046722, 225161564, 376201696, 262869988, 145798460, 262869988, 376201696, 225161564, 43046722},
{1000000001, 5937430213, 11798197840, 10137490792, 4649009794, 4649009794, 10137490792, 11798197840, 5937430213, 1000000001}
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MATHEMATICA
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Clear[p, x, n, m];
p[x_, n_] := ((1 - x)^(2*n)/(n*x))*Sum[Binomial[k + n - 1, k]*k^n*x^k, {k, 0, Infinity}];
Table[(CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x] + Reverse[ CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x]]), {n, 1, 10}];
Flatten[%]
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CROSSREFS
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Sequence in context: A123027 A100652 A094416 this_sequence A117030 A155758 A009097
Adjacent sequences: A152297 A152298 A152299 this_sequence A152301 A152302 A152303
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KEYWORD
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nonn
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AUTHOR
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Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Dec 02 2008
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