A154338 A triangular sequence of coefficients of polynomials: p(x,n)=(-(x - 1)^(n)*Sum[(((-1)^(n)*(2*k + 1)^(n - 1)))*x^k, {k,0, Infinity}]+2*(x - 1)^(n + 1)*Sum[((-1)^(n + 1)*k^n)*x^k, {k, 0, Infinity}]/x).

1, 1, 1, 1, 2, 1, 1, -1, -1, 1, 1, -24, -98, -24, 1, 1, -123, -1078, -1078, -123, 1, 1, -482, -8161, -18716, -8161, -482, 1, 1, -1685, -52071, -228485, -228485, -52071, -1685, 1, 1, -5548, -302396, -2308820, -4362634, -2308820, -302396, -5548, 1, 1, -17647, -1660660, -20797588, -66792586, -66792586, -20797588, -1660660, -17647, 1

Offset

0,5

Comments

Row sums are: {1, 2, 4, 0, -144, -2400, -36000, -564480, -9596160, -178536960,...}

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows

Formula

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

Functional form:

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

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

Example

{1},
{1, 1},
{1, 2, 1},
{1, -1, -1, 1},
{1, -24, -98, -24, 1},
{1, -123, -1078, -1078, -123, 1},
{1, -482, -8161, -18716, -8161, -482,1},
{1, -1685, -52071, -228485, -228485, -52071, -1685, 1},
{1, -5548, -302396, -2308820, -4362634, -2308820, -302396, -5548, 1},
{1, -17647, -1660660, -20797588, -66792586, -66792586, -20797588, -1660660, -17647, 1}

Mathematica

p[x_, n_] = (-(-1)^n*2^(-1 + n)*(-1 + x)^n*LerchPhi[x, 1 - n, 1/2] + 2*(-1)^(1 + n)*(-1 + x)^(1 + n)*PolyLog[-n, x]/x); Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 1, 50}]//Flatten (* G. C. Greubel, Sep 11 2016 *)

Crossrefs

Sequence in context: A155869 A176564 A237717 * A087436 A340831 A334862

Adjacent sequences: A154335 A154336 A154337 * A154339 A154340 A154341

Keyword

sign,tabl

Author

Roger L. Bagula, Jan 07 2009

Status

approved