A154702 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, 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

Offset: 3

Author

Roger L. Bagula, Jan 14 2009

Keywords

• nonn
• tabl

Comments

Row sums equal A037960(n+1) = (n + 2)!*n*(3*n + 1)/24.

Examples

			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;
		

Links

• G. C. Greubel, Rows n = 3..30 of triangle, flattened
• Roger L. Bagula, Fractal plot modulo two Mathematica code

Crossrefs

Cf. A037960.

Programs

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 *)

Extensions

Edited by G. C. Greubel, May 08 2019