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A154702 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. 0
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; refs; listen; history; text; internal format)
OFFSET

3,2

COMMENTS

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]

LINKS

Table of n, a(n) for n=3..38.

FORMULA

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.

EXAMPLE

{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}

MATHEMATICA

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[%]

CROSSREFS

A037960

Sequence in context: A121210 A241866 A243123 * A112685 A201958 A153721

Adjacent sequences:  A154699 A154700 A154701 * A154703 A154704 A154705

KEYWORD

nonn,uned

AUTHOR

Roger L. Bagula, Jan 14 2009

STATUS

approved

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Last modified January 18 20:57 EST 2019. Contains 319282 sequences. (Running on oeis4.)