A154337 A triangular sequence of coefficients of polynomials: p(x,n)=(3*(x - 1)^(n)*Sum[(((-1)^(n)*(2*k + 1)^(n - 1)))*x^k, {k,0, Infinity}] -(x - 1)^(n + 1)*Sum[((-1)^(n + 1)*k^n)*x^k, {k, 0, Infinity}]/x)/2.

1, 1, 1, 1, 7, 1, 1, 29, 29, 1, 1, 101, 312, 101, 1, 1, 327, 2372, 2372, 327, 1, 1, 1023, 15219, 34114, 15219, 1023, 1, 1, 3145, 88839, 381775, 381775, 88839, 3145, 1, 1, 9577, 490114, 3683815, 6934426, 3683815, 490114, 9577, 1, 1, 29003, 2610590, 32334362, 103464764, 103464764, 32334362, 2610590, 29003, 1

Offset: 0

Author

Roger L. Bagula, Jan 07 2009

Keywords

• nonn
• tabl

Comments

Row sums are: {1, 2, 9, 60, 516, 5400, 66600, 947520, 15301440, 276877440,...}

Examples

			{1},
{1, 1},
{1, 7, 1},
{1, 29, 29, 1},
{1, 101, 312, 101, 1},
{1, 327, 2372, 2372, 327, 1},
{1, 1023, 15219, 34114, 15219, 1023, 1},
{1, 3145, 88839, 381775, 381775, 88839, 3145, 1},
{1, 9577, 490114, 3683815, 6934426, 3683815, 490114, 9577, 1},
{1, 29003, 2610590, 32334362, 103464764, 103464764, 32334362, 2610590, 29003, 1}
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1274

Programs

Mathematica

Clear[p, x, n]; p[x_, n_] = (3*(x - 1)^(n)*Sum[(((-1)^(n)*(2*k + 1)^(n - 1)))*x^k, {k, 0, Infinity}]
- (x - 1)^(n + 1)*Sum[((-1)^(n + 1)*k^n)*x^k, {k, 0,Infinity}]/x)/2;
Table[FullSimplify[ExpandAll[p[x, n]]], {n, 1, 10}];
Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 1, 10}];
Flatten[%]

Formula

p(x,n)=(3*(x - 1)^(n)*Sum[(((-1)^(n)*(2*k + 1)^(n - 1)))*x^k, {k,0, Infinity}] -(x - 1)^(n + 1)*Sum[((-1)^(n + 1)*k^n)*x^k, {k, 0, Infinity}]/x)/2.

Functional form:

p(x,n)=(3*(-1)^n* 2^(-1 + n)* (-1 + x)^n* LerchPhi(x, 1 - n, 1/2) - (-1)^(1 + n) *(-1 + x)^(1 + n)* PolyLog( -n, x)/x)/2.

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