A155952 A triangle of polynomial coefficients: q(x,n)=-((x - 1)^(2*n + 1)/x^n)*Sum[(2*k + n)^n*Binomial[k, n]*x^k, {k, 0, Infinity}]; p(x,n)=q(x,n)+x^n*q(1/x,n).

2, 4, 4, 48, 48, 728, 232, 232, 728, 20752, 5312, 1632, 5312, 20752, 759132, 168684, 39864, 39864, 168684, 759132, 34016320, 5788288, 3904448, -2262272, 3904448, 5788288, 34016320, 1801010416, 223429840, 253864944, -64253360, -64253360

Offset

0,1

Comments

Row sums are:

{2, 8, 96, 1920, 53760, 1935360, 85155840, 4428103680, 265686220800,

18066663014400, 1373066389094400,...}.

Links

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

Formula

q(x,n)=-((x - 1)^(2*n + 1)/x^n)*Sum[(2*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, 4},
{48, 48},
{728, 232, 232, 728},
{20752, 5312, 1632, 5312, 20752},
{759132, 168684, 39864, 39864, 168684, 759132},
{34016320, 5788288, 3904448, -2262272, 3904448, 5788288, 34016320},
{1801010416, 223429840, 253864944, -64253360, -64253360, 253864944, 223429840, 1801010416},
{110076993792, 8135276544, 21010185216, -9977444352, 7196198400, -9977444352, 21010185216, 8135276544, 110076993792},
{7625557131380, 185854731220, 1792122898960, -827150318000, 256947063640, 256947063640, -827150318000, 1792122898960, 185854731220, 7625557131380},
{590491073741824, -15412908181504, 169164874601472, -90458315169792, 50709230659584, -35921522208768, 50709230659584, -90458315169792, 169164874601472, -15412908181504, 590491073741824}

Mathematica

Clear[p, x, n, m];
p[x_, n_] = -((x - 1)^(2*n + 1)/x^n)*Sum[(2*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: A320600 A360685 A290606 * A277445 A145636 A334190

Adjacent sequences: A155949 A155950 A155951 * A155953 A155954 A155955

Keyword

sign,tabl,uned

Author

Roger L. Bagula, Jan 31 2009

Status

approved