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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).
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%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