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A154336 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).

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