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A137497
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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].
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0
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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COMMENTS
| Row sums: {0, 0, 1, 3, 6, 0, -60, 0, 3360, 0, -544320};
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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)).
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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}
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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]
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CROSSREFS
| Sequence in context: A178514 A168426 A065931 * A032338 A081814 A133340
Adjacent sequences: A137494 A137495 A137496 * A137498 A137499 A137500
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KEYWORD
| uned,tabl,sign
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 22 2008
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