1,1

Row sums are: 2*n!;

{2, 2, 4, 12, 48, 240, 1440, 10080, 80640, 725760,...}.

It was very difficult to separate out the polynomial from the Log and PolyLog terms.

This term:

(-1)^n*n *(-1 + x)^(n - 1) Log[1 - x];

is very strange.

General polynomials based on sums of the sort:

Sum[(k + 1)^n*x^k/k^m, {k, 1, Infinity}];m=Integer

that are Zeta[m] like probably exist.

q(x,n)=-((-1)^n*(Sum[(k + 1)^n*x^k/k^2, {k, 1, Infinity}] - PolyLog[2, x])*(x - 1)^(n - 1)

+ (-1)^n*n *(-1 + x)^(n - 1) Log[1 - x])/x;

p(x,n)=q(x,n)+x^n*q(1/x,n);

t(n,m)=coefficients(p(x,n)).

{2},

{1, 1},

{17, -30, 17},

{16, -10, -10, 16},

{72, -176, 256, -176, 72},

{99, -57, 78, 78, -57, 99},

{275, -282, 1557, -1660, 1557, -282, 275},

{466, 1180, 2904, 490, 490, 2904, 1180, 466}, {1058, 5244, 21704, 4580, 15468, 4580, 21704, 5244, 1058}

Clear[p, x, n];

p[x_, n_] = -((-1)^n*(Sum[(k + 1)^n*x^k/k^2, {k, 1, Infinity}] - PolyLog[2, x])*(x - 1)^(n - 1) + (-1)^n*n *(-1 + x)^(n - 1) Log[1 - x])/x;

Table[FullSimplify[ExpandAll[p[x, n]]], {n, 1, 10}];

Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x] + Reverse[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x]], {n, 1, 10}];

Flatten[%]

Sequence in context: A078089 A095836 A156697 this_sequence A090163 A124001 A157453

Adjacent sequences: A154988 A154989 A154990 this_sequence A154992 A154993 A154994

sign,tabl,uned

Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 18 2009