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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
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
3,2
COMMENTS
Row sums equal A037960(n+1) = (n + 2)!*n*(3*n + 1)/24.
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: A241866 A243123 A372230 * A112685 A201958 A351477
KEYWORD
nonn
AUTHOR
Roger L. Bagula, Jan 14 2009
EXTENSIONS
Edited by G. C. Greubel, May 08 2019
STATUS
approved