A155947 A triangle of polynomial coefficients: q(x,n)=(1 - x)^(n + 1)*Sum[(k + n)^n*x^k, {k, 0, Infinity}]; p(x,n)=q(x,n)+x^n*q(1/x,n).

1, 1, 2, 5, -6, 5, 19, -13, -13, 19, 337, -1044, 1462, -1044, 337, 2101, -5073, 3092, 3092, -5073, 2101, 62281, -314222, 718559, -931796, 718559, -314222, 62281, 543607, -2329829, 3835365, -2044103, -2044103, 3835365, -2329829, 543607

Offset

0,3

Comments

Row sums are:2*n!

{2, 2, 4, 12, 48, 240, 1440, 10080, 80640, 725760, 7257600,...}.

The result is related to the Eulerian numbers infinite sum form.

This was the result of finding the infinite sum identity:

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

Links

Table of n, a(n) for n=0..35.

Formula

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

q(x,n)=(1 - x)^(n + 1)*LerchPhi[x, -n, n];

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

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

Example

{1, 1},
{2},
{5, -6, 5},
{19, -13, -13, 19},
{337, -1044, 1462, -1044, 337},
{2101, -5073, 3092, 3092, -5073, 2101},
{62281, -314222, 718559, -931796, 718559, -314222, 62281},
{543607, -2329829, 3835365, -2044103, -2044103, 3835365, -2329829, 543607},
{22542017, -158151816, 509366204, -972472504, 1197512838, -972472504, 509366204, -158151816, 22542017},
{253202761, -1572381217, 4145530310, -5521116358, 2695127384, 2695127384, -5521116358, 4145530310, -1572381217, 253202761},
{13486784401, -121343461986, 506850150853, -1285984548968, 2186943445546, -2599897482092, 2186943445546, -1285984548968, 506850150853, -121343461986, 13486784401}

Mathematica

Clear[p, x, n, m];
p[x_, n_] = (1 - x)^(n + 1)*Sum[(k + n)^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: A161017 A198231 A272207 * A008294 A019694 A233588

Adjacent sequences: A155944 A155945 A155946 * A155948 A155949 A155950

Keyword

sign,tabl,uned

Author

Roger L. Bagula, Jan 31 2009

Status

approved