A137499 A triangular sequence of coefficients from a Laplace Transform of a Bernoulli expansion function: LaplaceTransform[t*Exp[x*t]/(Exp[t] - 1), t, 1/t] = Zeta[2,1+1/t-x]->shifted to Zeta[6,1+1/t-x].

0, 0, 0, 0, 0, 24, -360, 720, 2520, -15120, 15120, 0, 141120, -423360, 282240, -169344, 0, 5080320, -10160640, 5080320, 0, -15240960, 0, 152409600, -228614400, 91445760

Offset

1,6

Comments

Row sums: {0, 0, 0, 0, 0, 24, 360, 2520, 0, -169344, 0};

These functions are due to the close connection of the Bernoulli-type functions with the Zeta (generalized) functions.

Links

Table of n, a(n) for n=1..26.

Formula

Zeta[6,1+1/t-x] = Sum[1/(n+1/t+x)^6,{n,0,Infinity}] = Sum[p(x,n)*t^n/n!,{n,0,Infinity}]; out(n,m)=n!*Coefficients(p(x,n)).

Example

{0},
{0},
{0},
{0},
{0},
{24},
{-360,720},
{2520, -15120, 15120},
{0, 141120, -423360, 282240},
{-169344, 0, 5080320, -10160640, 5080320},
{0, -15240960, 0, 152409600, -228614400, 91445760}

Mathematica

LaplaceTransform[t*Exp[x*t]/(Exp[t] - 1), t, s]; Clear[p, f, g] p[t_] = Zeta[6, 1 + 1/t - x]; Table[ ExpandAll[n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a = Table[ CoefficientList[n!*SeriesCoefficient[ FullSimplify[Series[p[t], {t, 0, 30}]], n], x], {n, 0, 10}]; Flatten[a]

Crossrefs

Sequence in context: A004325 A075621 A293618 * A371068 A122813 A028245

Adjacent sequences: A137496 A137497 A137498 * A137500 A137501 A137502

Keyword

uned,tabf,sign

Author

Roger L. Bagula, Apr 22 2008

Status

approved