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A154702 Second derivative of Eulerian number polynomials as a triangular sequence defined by T(n, m) = Coefficients(q(x,n) + x^(n-2)*q(1/x,n))/4, where q(x, n) = d^2*P(x, n)/dx^2 and p(x, n)=(x-1)^(n+1)*Sum_{k>=0} ((-1)^(n + 1)*k^n)*x^(k-1). 1
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 equal A037960(n+1) = (n + 2)!*n*(3*n + 1)/24 = {1, 14, 150, 1560, 16800, 191520, 2328480, 30240000, ...}.

LINKS

G. C. Greubel, Rows n = 3..30 of triangle, flattened

Roger L. Bagula, Fractal plot modulo two Mathematica code

FORMULA

Triangle sequence generated by T(n, m) = Coefficients(q(x,n) + x^(n-2)*q(1/x,n))/4, where q(x, n) = d^2*P(x, n)/dx^2 and p(x, n)=(x-1)^(n+1)*Sum_{k>=0} ((-1)^(n + 1)*k^n)*x^(k-1).

EXAMPLE

Triangle begins as:

      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

p[x_, n_] := Sum[k!*StirlingS2[n, k]*(x - 1)^(n - k), {k, 1, n}];

(* or 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}];

f[n_]:= CoefficientList[FullSimplify[ExpandAll[q[x, n]]], x];

Table[(f[n] + Reverse[f[n]])/4, {n, 1, 10}]//Flatten (* modified by G. C. Greubel, May 08 2019 *)

CROSSREFS

Cf. A037960.

Sequence in context: A121210 A241866 A243123 * A112685 A201958 A153721

Adjacent sequences:  A154699 A154700 A154701 * A154703 A154704 A154705

KEYWORD

nonn

AUTHOR

Roger L. Bagula, Jan 14 2009

EXTENSIONS

Edited by G. C. Greubel, May 08 2019

STATUS

approved

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Last modified July 10 23:30 EDT 2020. Contains 335600 sequences. (Running on oeis4.)