A137497 Triangular sequence of coefficients from the 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[3,1+1/t-x].

0, 0, 1, -3, 6, 6, -36, 36, 0, 120, -360, 240, -60, 0, 1800, -3600, 1800, 0, -2520, 0, 25200, -37800, 15120, 3360, 0, -70560, 0, 352800, -423360, 141120, 0, 241920, 0, -1693440, 0, 5080320, -5080320, 1451520, -544320, 0, 10886400, 0, -38102400, 0, 76204800, -65318400, 16329600

Offset

1,4

Comments

Row sums: {0, 0, 1, 3, 6, 0, -60, 0, 3360, 0, -544320};

Links

Table of n, a(n) for n=1..47.

Formula

Zeta[3,1+1/t-x]=Sum[1/(n+1/t+x)^3,{n,0,Infinity}]=Sum[p(x,n)*t^n/n!,{n,0,Infinity}]; out(n,m)=n!*Coefficients(p(x,n)).

Example

{0},
{0},
{1},
{-3, 6},
{6, -36, 36},
{0, 120, -360, 240},
{-60, 0, 1800, -3600, 1800},
{0, -2520, 0, 25200, -37800, 15120},
{3360, 0, -70560, 0, 352800, -423360, 141120},
{0, 241920, 0, -1693440, 0, 5080320, -5080320,1451520},
{-544320, 0, 10886400, 0, -38102400, 0,76204800, -65318400, 16329600}

Mathematica

LaplaceTransform[t*Exp[x*t]/(Exp[t] - 1), t, s]; Clear[p, f, g] p[t_] = Zeta[3, 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: A181372 A168426 A065931 * A032338 A081814 A133340

Adjacent sequences: A137494 A137495 A137496 * A137498 A137499 A137500

Keyword

uned,tabl,sign

Author

Roger L. Bagula, Apr 22 2008

Status

approved