A154989 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).

2, 1, 1, 10, -16, 10, 11, -5, -5, 11, 36, -58, 92, -58, 36, 57, 21, 42, 42, 21, 57, 134, 156, 618, -376, 618, 156, 134, 247, 1303, 2529, 961, 961, 2529, 1303, 247, 520, 5162, 17524, 12646, 8936, 12646, 17524, 5162, 520, 1013, 19393, 99880, 153472, 89122

Offset

1,1

Comments

Row sums are: 2*n!;

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

Links

Table of n, a(n) for n=1..50.

Formula

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);

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

Example

{2},
{1, 1},
{10, -16, 10},
{11, -5, -5, 11},
{36, -58, 92, -58, 36},
{57, 21, 42, 42, 21, 57},
{134, 156, 618, -376, 618, 156, 134},
{247, 1303, 2529, 961, 961, 2529, 1303, 247},
{520, 5162, 17524, 12646, 8936, 12646, 17524, 5162, 520},
{1013, 19393, 99880, 153472, 89122, 89122, 153472, 99880, 19393, 1013}

Mathematica

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

Crossrefs

Sequence in context: A153731 A262226 A298158 * A064307 A165883 A370950

Adjacent sequences: A154986 A154987 A154988 * A154990 A154991 A154992

Keyword

uned,sign

Author

Roger L. Bagula, Jan 18 2009

Status

approved