A155950 A triangle of polynomial coefficients: q(x,n)=(1 - x)^(n + 1)*Sum[(2*k + n)^n*x^k, {k, 0, Infinity}]; p(x,n)=q(x,n)+x^n*q(1/x,n).

1, 1, 2, 2, 8, 8, 26, 22, 22, 26, 272, -64, 352, -64, 272, 2882, -486, 1444, 1444, -486, 2882, 50752, -93056, 230336, -283904, 230336, -93056, 50752, 745418, -1182562, 2112618, -1030354, -1030354, 2112618, -1182562, 745418, 18456832, -66045952

Offset

0,3

Comments

Row sums are:

{2, 4, 16, 96, 768, 7680, 92160, 1290240, 20643840, 371589120, 7431782400,...}.

Links

Table of n, a(n) for n=0..37.

Formula

q(x,n)=(1 - x)^(n + 1)*Sum[(2*k + n)^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, 1},
{2, 2},
{8, 8},
{26, 22, 22, 26},
{272, -64, 352, -64, 272},
{2882, -486, 1444, 1444, -486, 2882}, {50752, -93056, 230336, -283904, 230336, -93056, 50752},
{745418, -1182562, 2112618, -1030354, -1030354, 2112618, -1182562, 745418},
{18456832, -66045952, 193838080, -342063104, 412272128, -342063104, 193838080, -66045952, 18456832},
{347066882, -1114674254, 2662543720, -3229707896, 1520566108, 1520566108, -3229707896, 2662543720, -1114674254, 347066882},
{11073741824, -59833329664, 216555369472, -500687839232, 812895791104, -952575684608, 812895791104, -500687839232, 216555369472, -59833329664, 11073741824}

Mathematica

Clear[p, x, n, m];
p[x_, n_] = (1 - x)^(n + 1)*Sum[(2*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: A099328 A073090 A120544 * A162959 A354830 A158302

Adjacent sequences: A155947 A155948 A155949 * A155951 A155952 A155953

Keyword

sign,tabl,uned

Author

Roger L. Bagula, Jan 31 2009

Status

approved