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A137496
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Irregular triangle read by rows: coefficients of Laplace transform of a Bernoulli expansion: LaplaceTransform[t*Exp[x*t]/(Exp[t] - 1), t, 1/t] = Zeta[2,1+1/t-x]->shifted to Zeta[4,1+1/t-x].
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0
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0, 0, 0, 2, -12, 24, 40, -240, 240, 0, 1200, -3600, 2400, -840, 0, 25200, -50400, 25200, 0, -47040, 0, 470400, -705600, 282240, 80640, 0, -1693440, 0, 8467200, -10160640, 3386880, 0, 7257600, 0, -50803200, 0, 152409600, -152409600, 43545600
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OFFSET
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1,4
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COMMENTS
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Row sums: {0, 0, 0, 2, 12, 40, 0, -840, 0, 80640, 0, ...}.
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LINKS
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FORMULA
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Zeta[4,1+1/t-x]=Sum[1/(n+1/t+x)^4,{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
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{0},
{0},
{0},
{2},
{-12, 24},
{40, -240, 240},
{0, 1200, -3600, 2400},
{-840, 0, 25200, -50400, 25200},
{0, -47040,0, 470400, -705600, 282240},
{80640, 0, -1693440, 0, 8467200, -10160640, 3386880},
{0, 7257600, 0, -50803200, 0, 152409600, -152409600, 43545600}
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MATHEMATICA
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LaplaceTransform[t*Exp[x*t]/(Exp[t] - 1), t, s]; Clear[p, f, g] p[t_] = Zeta[4, 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
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KEYWORD
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tabf,sign
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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