A154989 - OEIS

%I #3 Mar 30 2012 17:34:32

%S 2,1,1,10,-16,10,11,-5,-5,11,36,-58,92,-58,36,57,21,42,42,21,57,134,

%T 156,618,-376,618,156,134,247,1303,2529,961,961,2529,1303,247,520,

%U 5162,17524,12646,8936,12646,17524,5162,520,1013,19393,99880,153472,89122

%N Symmetrical triangle sequence from polynomials: q(x,n)=(-1)^n*(Sum[(k + 1)^n*x^k/k, {k, 1, Infinity}] + Log[1 - x])*(x - 1)^n/x; p(x,n)=q(x,n)+x^n*q(1/x,n).

%C Row sums are: 2*n!;

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

%F q(x,n)=(-1)^n*(Sum[(k + 1)^n*x^k/k, {k, 1, Infinity}] + Log[1 - x])*(x - 1)^n/x;

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

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

%e {2},

%e {1, 1},

%e {10, -16, 10},

%e {11, -5, -5, 11},

%e {36, -58, 92, -58, 36},

%e {57, 21, 42, 42, 21, 57},

%e {134, 156, 618, -376, 618, 156, 134},

%e {247, 1303, 2529, 961, 961, 2529, 1303, 247},

%e {520, 5162, 17524, 12646, 8936, 12646, 17524, 5162, 520},

%e {1013, 19393, 99880, 153472, 89122, 89122, 153472, 99880, 19393, 1013}

%t Clear[p, x, n];

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

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

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

%t Flatten[%]

%K uned,sign

%O 1,1

%A _Roger L. Bagula_, Jan 18 2009