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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).

%I A154702 #14 Feb 19 2025 17:31:35
%S A154702 1,7,7,36,78,36,156,624,624,156,603,4224,7146,4224,603,2157,25281,
%T A154702 68322,68322,25281,2157,7318,137622,578130,882340,578130,137622,7318,
%U A154702 23938,696970,4433382,9965710,9965710,4433382,696970,23938
%N 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).
%C A154702 Row sums equal A037960(n+1) = (n + 2)!*n*(3*n + 1)/24.
%H A154702 G. C. Greubel, <a href="/A154702/b154702.txt">Rows n = 3..30 of triangle, flattened</a>
%H A154702 Roger L. Bagula, <a href="/A154702/a154702.txt">Fractal plot modulo two Mathematica code</a>
%e A154702 Triangle begins as:
%e A154702       1;
%e A154702       7,      7;
%e A154702      36,     78,      36;
%e A154702     156,    624,     624,     156;
%e A154702     603,   4224,    7146,    4224,     603;
%e A154702    2157,  25281,   68322,   68322,   25281,    2157;
%e A154702    7318, 137622,  578130,  882340,  578130,  137622,   7318;
%e A154702   23938, 696970, 4433382, 9965710, 9965710, 4433382, 696970, 23938;
%t A154702 p[x_, n_] := Sum[k!*StirlingS2[n, k]*(x - 1)^(n - k), {k, 1, n}];
%t A154702 (* or p[x_, n_]:= (x-1)^(n+1)*Sum[((-1)^(n+1)*k^n)*x^k, {k, 0, Infinity}]/x; *)
%t A154702 q[x_, n_]:= D[p[x, n], {x, 2}];
%t A154702 f[n_]:= CoefficientList[FullSimplify[ExpandAll[q[x, n]]], x];
%t A154702 Table[(f[n] + Reverse[f[n]])/4, {n, 1, 10}]//Flatten (* modified by _G. C. Greubel_, May 08 2019 *)
%Y A154702 Cf. A037960.
%K A154702 nonn,tabl
%O A154702 3,2
%A A154702 _Roger L. Bagula_, Jan 14 2009
%E A154702 Edited by _G. C. Greubel_, May 08 2019