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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.
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%I #10 Sep 13 2016 07:58:54

%S 1,1,1,1,7,1,1,29,29,1,1,101,312,101,1,1,327,2372,2372,327,1,1,1023,

%T 15219,34114,15219,1023,1,1,3145,88839,381775,381775,88839,3145,1,1,

%U 9577,490114,3683815,6934426,3683815,490114,9577,1,1,29003,2610590,32334362,103464764,103464764,32334362,2610590,29003,1

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

%C Row sums are: {1, 2, 9, 60, 516, 5400, 66600, 947520, 15301440, 276877440,...}

%H G. C. Greubel, <a href="/A154337/b154337.txt">Table of n, a(n) for n = 0..1274</a>

%F 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 Functional form:

%F 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 t(n,m)=Coefficients(p(x,n))

%e {1},

%e {1, 1},

%e {1, 7, 1},

%e {1, 29, 29, 1},

%e {1, 101, 312, 101, 1},

%e {1, 327, 2372, 2372, 327, 1},

%e {1, 1023, 15219, 34114, 15219, 1023, 1},

%e {1, 3145, 88839, 381775, 381775, 88839, 3145, 1},

%e {1, 9577, 490114, 3683815, 6934426, 3683815, 490114, 9577, 1},

%e {1, 29003, 2610590, 32334362, 103464764, 103464764, 32334362, 2610590, 29003, 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}]

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

%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