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A154337 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}] -(x - 1)^(n + 1)*Sum[((-1)^(n + 1)*k^n)*x^k, {k, 0, Infinity}]/x)/2.

%I A154337 #10 Sep 13 2016 07:58:54
%S A154337 1,1,1,1,7,1,1,29,29,1,1,101,312,101,1,1,327,2372,2372,327,1,1,1023,
%T A154337 15219,34114,15219,1023,1,1,3145,88839,381775,381775,88839,3145,1,1,
%U A154337 9577,490114,3683815,6934426,3683815,490114,9577,1,1,29003,2610590,32334362,103464764,103464764,32334362,2610590,29003,1
%N A154337 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}] -(x - 1)^(n + 1)*Sum[((-1)^(n + 1)*k^n)*x^k, {k, 0, Infinity}]/x)/2.
%C A154337 Row sums are: {1, 2, 9, 60, 516, 5400, 66600, 947520, 15301440, 276877440,...}
%H A154337 G. C. Greubel, <a href="/A154337/b154337.txt">Table of n, a(n) for n = 0..1274</a>
%F A154337 p(x,n)=(3*(x - 1)^(n)*Sum[(((-1)^(n)*(2*k + 1)^(n - 1)))*x^k, {k,0, Infinity}] -(x - 1)^(n + 1)*Sum[((-1)^(n + 1)*k^n)*x^k, {k, 0, Infinity}]/x)/2.
%F A154337 Functional form:
%F A154337 p(x,n)=(3*(-1)^n* 2^(-1 + n)* (-1 + x)^n* LerchPhi(x, 1 - n, 1/2) - (-1)^(1 + n) *(-1 + x)^(1 + n)* PolyLog( -n, x)/x)/2.
%F A154337 t(n,m)=Coefficients(p(x,n))
%e A154337 {1},
%e A154337 {1, 1},
%e A154337 {1, 7, 1},
%e A154337 {1, 29, 29, 1},
%e A154337 {1, 101, 312, 101, 1},
%e A154337 {1, 327, 2372, 2372, 327, 1},
%e A154337 {1, 1023, 15219, 34114, 15219, 1023, 1},
%e A154337 {1, 3145, 88839, 381775, 381775, 88839, 3145, 1},
%e A154337 {1, 9577, 490114, 3683815, 6934426, 3683815, 490114, 9577, 1},
%e A154337 {1, 29003, 2610590, 32334362, 103464764, 103464764, 32334362, 2610590, 29003, 1}
%t A154337 Clear[p, x, n]; p[x_, n_] = (3*(x - 1)^(n)*Sum[(((-1)^(n)*(2*k + 1)^(n - 1)))*x^k, {k, 0, Infinity}]
%t A154337 - (x - 1)^(n + 1)*Sum[((-1)^(n + 1)*k^n)*x^k, {k, 0,Infinity}]/x)/2;
%t A154337 Table[FullSimplify[ExpandAll[p[x, n]]], {n, 1, 10}];
%t A154337 Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 1, 10}];
%t A154337 Flatten[%]
%K A154337 nonn,tabl
%O A154337 0,5
%A A154337 _Roger L. Bagula_, Jan 07 2009