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%I #10 Sep 16 2016 11:19:44
%S 1,1,1,1,10,1,1,47,47,1,1,176,558,176,1,1,597,4442,4442,597,1,1,1926,
%T 29247,65812,29247,1926,1,1,6043,173385,747931,747931,173385,6043,1,1,
%U 18652,965620,7279396,13712662,7279396,965620,18652,1,1,56993,5173340,64213532,205619174,205619174,64213532,5173340,56993,1
%N A triangular sequence of coefficients of polynomials: p(x,n)=(3*(x - 1)^(n)*Sum[(((-1)^(n)*(2*k + 1)^(n - 1)))*x^k, {k,0, Infinity}] -2*(x - 1)^(n + 1)*Sum[((-1)^(n + 1)*k^n)*x^k, {k, 0, Infinity}]/x).
%C Row sums are: {1, 2, 12, 96, 912, 10080, 128160, 1854720, 30240000, 550126080,...}
%H G. C. Greubel, <a href="/A154336/b154336.txt">Table of n, a(n) for the first 50 rows</a>
%F p(x,n)=(3*(x - 1)^(n)*Sum[(((-1)^(n)*(2*k + 1)^(n - 1)))*x^k, {k,0, Infinity}] -2*(x - 1)^(n + 1)*Sum[((-1)^(n + 1)*k^n)*x^k, {k, 0, Infinity}]/x).
%F Functional form:
%F p(x,n)=(3*(-1)^n* 2^(-1 + n)* (-1 + x)^n* LerchPhi(x, 1 - n, 1/2) - 2*(-1)^(1 + n) *(-1 + x)^(1 + n)* PolyLog( -n, x)/x).
%F t(n,m)=Coefficients(p(x,n))
%e {1},
%e {1, 1},
%e {1, 10, 1},
%e {1, 47, 47, 1},
%e {1, 176, 558, 176, 1},
%e {1, 597, 4442, 4442, 597, 1},
%e {1, 1926, 29247, 65812, 29247, 1926, 1},
%e {1, 6043, 173385, 747931, 747931, 173385, 6043, 1},
%e {1, 18652, 965620, 7279396, 13712662, 7279396, 965620, 18652, 1},
%e {1, 56993, 5173340, 64213532, 205619174, 205619174, 64213532, 5173340, 56993, 1}
%t Clear[p, x, n]; p[x_, n_] = (3*(x - 1)^(n)*Sum[(((-1)^(n)*(2*k + 1)^(n - 1)))*x^k, {k, 0, Infinity}] - 2*(x - 1)^(n + 1)*Sum[((-1)^(n + 1)*k^n) * x^k, {k, 0,Infinity}]/x);
%t Table[FullSimplify[ExpandAll[p[x, n]]], {n, 1, 10}];
%t Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 1, 10}];
%t Flatten[%]
%K nonn,tabl
%O 0,5
%A _Roger L. Bagula_, Jan 07 2009
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