A155951 Triangle read by rows. Let q(x,n) = -((x - 1)^(2*n + 1)/x^n)*Sum[(k + n)^n*Binomial[k, n]*x^k, {k, 0, Infinity}]; p(x,n)=q(x,n)+x^n*q(1/x,n); then row n gives coefficients of p(x,n).

2, 4, 17, -10, 17, 208, -88, -88, 208, 4177, -4708, 4422, -4708, 4177, 98976, -123888, 55152, 55152, -123888, 98976, 3001609, -5204582, 5360567, -4984628, 5360567, -5204582, 3001609, 105133568, -210753520, 208361232, -85444000, -85444000

Offset

0,1

Comments

Row sums are in A151817.

Links

Table of n, a(n) for n=0..31.

Formula

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

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

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

Example

{2},
{4},
{17, -10, 17},
{208, -88, -88, 208},
{4177, -4708, 4422, -4708, 4177},
{98976, -123888, 55152, 55152, -123888, 98976},
{3001609, -5204582, 5360567, -4984628, 5360567, -5204582, 3001609},
{105133568, -210753520, 208361232, -85444000, -85444000, 208361232, -210753520, 105133568},
{4300732097, -10315512136, 13267499516, -12384821752, 11302041350, -12384821752, 13267499516, -10315512136, 4300732097},
{198225072640, -539802938440, 752937755480, -641425101400, 247708437320, 247708437320, -641425101400, 752937755480, -539802938440, 198225072640},
{10243486784401, -31622720552146, 50805231998853, -55277019174408, 48150459465066, -43257991897932, 48150459465066, -55277019174408, 50805231998853, -31622720552146, 10243486784401}

Mathematica

Clear[p, x, n, m];
p[x_, n_] = -((x - 1)^(2*n + 1)/x^n)*Sum[(k + n)^n*Binomial[k, n]*x^k, {k, 0, Infinity}];
Table[FullSimplify[ExpandAll[p[x, n]]], {n, 0, 10}];
Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x]
+ Reverse[ CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x]], {n, 0, 10}];
Flatten[%]

Crossrefs

Sequence in context: A249760 A271552 A105510 * A208132 A254206 A118242

Adjacent sequences: A155948 A155949 A155950 * A155952 A155953 A155954

Keyword

sign,tabf

Author

Roger L. Bagula, Jan 31 2009

Extensions

Edited by N. J. A. Sloane, Jul 05 2009

Status

approved