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Search: id:A154702
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| A154702 |
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Second derivative of Eulerian number polynomials as a symmetrical triangular sequence: p(x,n)=(x - 1)^(n + 1)*Sum[((-1)^(n + 1)*k^n)*x^k, {k, 0, Infinity}]/x; q(x,n)=d^2*P(x,n)/dx^2; t(n,m)=Coefficients(q(x,n)+x^(n-2)*q(1/x,n))/4. |
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| 1, 7, 7, 36, 78, 36, 156, 624, 624, 156, 603, 4224, 7146, 4224, 603, 2157, 25281, 68322, 68322, 25281, 2157, 7318, 137622, 578130, 882340, 578130, 137622, 7318, 23938, 696970, 4433382, 9965710, 9965710, 4433382, 696970, 23938
(list; graph; listen)
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OFFSET
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3,2
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COMMENT
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Row sums are:A037960;(n + 2)!*n*(3*n + 1)/24
{1, 14, 150, 1560, 16800, 191520, 2328480, 30240000,...}.
The fractal plot modulo two is a dust:
a = Table[(CoefficientList[FullSimplify[ExpandAll[q[x,n]]], x] + Reverse[CoefficientList[FullSimplify[ExpandAll[q[x, n]]], x]])/4, {n, 3, 32}]; b = Table[If[m <= n, Mod[a[[n]][[m]], 2], 0], {m, 1, Length[a]}, {n, 1, Length[a]}];
ListDensityPlot[b, Mesh -> False]
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FORMULA
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p(x,n)=(x - 1)^(n + 1)*Sum[((-1)^(n + 1)*k^n)*x^k, {k, 0, Infinity}]/x;
q(x,n)=d^2*P(x,n)/dx^2;
t(n,m)=Coefficients(q(x,n)+x^(n-2)*q(1/x,n))/4.
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EXAMPLE
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{1},
{7, 7},
{36, 78, 36},
{156, 624, 624, 156},
{603, 4224, 7146, 4224, 603},
{2157, 25281, 68322, 68322, 25281, 2157},
{7318, 137622, 578130, 882340, 578130, 137622, 7318},
{23938, 696970, 4433382, 9965710, 9965710, 4433382, 696970, 23938}
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MATHEMATICA
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Clear[p, x, n]; p[x_, n_] = (x - 1)^(n + 1)*Sum[((-1)^(n + 1)*k^n)*x^k, {k, 0, Infinity}]/x;
q[x_, n_] = D[p[x, n], {x, 2}];
Table[(CoefficientList[FullSimplify[ExpandAll[q[x, n]]], x] + Reverse[ CoefficientList[FullSimplify[ExpandAll[q[x, n]]], x]])/4, {n, 1, 10}];
Flatten[%]
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CROSSREFS
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A037960
Sequence in context: A140252 A095343 A121210 this_sequence A112685 A153721 A151791
Adjacent sequences: A154699 A154700 A154701 this_sequence A154703 A154704 A154705
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KEYWORD
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nonn,uned
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AUTHOR
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Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 14 2009
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