A177984 A symmetrical triangle of polynomial coefficients:p(x,n)=If[n == 0, 1, (1 - x)^(n + 1)*Sum[((2*k + 1)^n + (k + 1)^n + k^n)*x^k, {k, 0, Infinity}]/2].

1, 1, 1, 1, 4, 1, 1, 14, 14, 1, 1, 44, 126, 44, 1, 1, 132, 887, 887, 132, 1, 1, 390, 5451, 12076, 5451, 390, 1, 1, 1150, 30984, 131665, 131665, 30984, 1150, 1, 1, 3400, 168076, 1252600, 2353126, 1252600, 168076, 3400, 1, 1, 10088, 885725, 10905407, 34828859

Offset: 0

Author

Roger L. Bagula, May 16 2010

Keywords

• nonn
• tabl
• uned

Comments

Row sums are:

{1, 2, 6, 30, 216, 2040, 23760, 327600, 5201280, 93260160, 1861574400,...}.

Examples

			{1},
{1, 1},
{1, 4, 1},
{1, 14, 14, 1},
{1, 44, 126, 44, 1},
{1, 132, 887, 887, 132, 1},
{1, 390, 5451, 12076, 5451, 390, 1},
{1, 1150, 30984, 131665, 131665, 30984, 1150, 1},
{1, 3400, 168076, 1252600, 2353126, 1252600, 168076, 3400, 1},
{1, 10088, 885725, 10905407, 34828859, 34828859, 10905407, 885725, 10088, 1},
{1, 30026, 4582497, 89401968, 454344414, 764856588, 454344414, 89401968, 4582497, 30026, 1}
		

Crossrefs

Cf. A008518, A060187

Programs

Mathematica

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

Formula

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

t(n,m)=coefficients(p(x,n))=If[n==0,1,(A008518(n,m)+A060187(n,m))/2]