Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.
%I #10 Jun 03 2023 06:29:08
%S 1,7,7,36,78,36,156,624,624,156,603,4224,7146,4224,603,2157,25281,
%T 68322,68322,25281,2157,7318,137622,578130,882340,578130,137622,7318,
%U 23938,696970,4433382,9965710,9965710,4433382,696970,23938
%N 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).
%C Row sums equal A037960(n+1) = (n + 2)!*n*(3*n + 1)/24.
%H G. C. Greubel, <a href="/A154702/b154702.txt">Rows n = 3..30 of triangle, flattened</a>
%H Roger L. Bagula, <a href="/A154702/a154702.txt">Fractal plot modulo two Mathematica code</a>
%e Triangle begins as:
%e 1;
%e 7, 7;
%e 36, 78, 36;
%e 156, 624, 624, 156;
%e 603, 4224, 7146, 4224, 603;
%e 2157, 25281, 68322, 68322, 25281, 2157;
%e 7318, 137622, 578130, 882340, 578130, 137622, 7318;
%e 23938, 696970, 4433382, 9965710, 9965710, 4433382, 696970, 23938;
%t p[x_, n_] := Sum[k!*StirlingS2[n, k]*(x - 1)^(n - k), {k, 1, n}];
%t (* or p[x_, n_]:= (x-1)^(n+1)*Sum[((-1)^(n+1)*k^n)*x^k, {k, 0, Infinity}]/x; *)
%t q[x_, n_]:= D[p[x, n], {x, 2}];
%t f[n_]:= CoefficientList[FullSimplify[ExpandAll[q[x, n]]], x];
%t Table[(f[n] + Reverse[f[n]])/4, {n, 1, 10}]//Flatten (* modified by _G. C. Greubel_, May 08 2019 *)
%Y Cf. A037960.
%K nonn
%O 3,2
%A _Roger L. Bagula_, Jan 14 2009
%E Edited by _G. C. Greubel_, May 08 2019