A155915 A triangle of polynomial coefficients: q(x,n)=-((x - 1)^(2*n + 1)/x^n)*Sum[(2*k + 1)^n*Binomial[k, n]*x^k, {k, 0, Infinity}]; p(x,n)=q(x,n)+x^n*q(1/x,n).

1, 2, 2, 13, 22, 13, 172, 308, 308, 172, 3281, 7276, 5766, 7276, 3281, 80526, 228822, 174492, 174492, 228822, 80526, 2413405, 8495474, 8083699, 4592764, 8083699, 8495474, 2413405, 85429688, 359918120, 440763192, 220914920, 220914920

Offset

0,2

Comments

Row sums are:

{1, 4, 48, 960, 26880, 967680, 42577920, 2214051840, 132843110400,

9033331507200, 686533194547200}

Links

Table of n, a(n) for n=0..32.

Formula

q(x,n)=-((x - 1)^(2*n + 1)/x^n)*Sum[(2*k + 1)^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

{1},
{2, 2},
{13, 22, 13},
{172, 308, 308, 172},
{3281, 7276, 5766, 7276, 3281},
{80526, 228822, 174492, 174492, 228822, 80526},
{2413405, 8495474, 8083699, 4592764, 8083699, 8495474, 2413405},
{85429688, 359918120, 440763192, 220914920, 220914920, 440763192, 359918120, 85429688},
{3487878721, 17132124952, 26131556188, 15925828264, 7488334150, 15925828264, 26131556188, 17132124952, 3487878721}, {161343848890, 905867202410, 1664943766280, 1285119074600, 499391861420, 499391861420, 1285119074600, 1664943766280, 905867202410, 161343848890},
{8339940489101, 52707061728718, 113751017120841, 108335058426024, 49780261735722, 20706515546388, 49780261735722, 108335058426024, 113751017120841, 52707061728718, 8339940489101}

Mathematica

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

Crossrefs

A155164

Sequence in context: A141575 A306738 A151352 * A173466 A151367 A368957

Adjacent sequences: A155912 A155913 A155914 * A155916 A155917 A155918

Keyword

nonn,tabl,uned

Author

Roger L. Bagula, Jan 30 2009

Status

approved