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%I A137498 #8 Dec 24 2018 21:40:43
%S A137498 0,0,0,0,6,-60,120,300,-1800,1800,0,12600,-37800,25200,-11760,0,
%T A137498 352800,-705600,352800,0,-846720,0,8467200,-12700800,5080320,1814400,
%U A137498 0,-38102400,0,190512000,-228614400,76204800
%N A137498 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[5,1+1/t-x].
%C A137498 Row sums: {0, 0, 0, 0, 6, 60, 300, 0, -11760, 0, 1814400};
%C A137498 These functions are due to the close connection of the Bernoulli-type functions with the Zeta (generalized) functions.
%F A137498 Zeta[5,1+1/t-x] = Sum[1/(n+1/t+x)^5,{n,0,Infinity}] = Sum[p(x,n)*t^n/n!,{n,0,Infinity}]; out(n,m)=n!*Coefficients(p(x,n)).
%e A137498 {0},
%e A137498 {0},
%e A137498 {0},
%e A137498 {0},
%e A137498 {6},
%e A137498 {-60, 120},
%e A137498 {300, -1800, 1800},
%e A137498 {0, 12600, -37800, 25200},
%e A137498 {-11760, 0, 352800, -705600, 352800},
%e A137498 {0, -846720, 0, 8467200, -12700800, 5080320},
%e A137498 {1814400, 0, -38102400, 0, 190512000, -228614400, 76204800}
%t A137498 LaplaceTransform[t*Exp[x*t]/(Exp[t] - 1), t, s]; Clear[p, f, g] p[t_] = Zeta[5, 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]
%K A137498 uned,tabf,sign
%O A137498 1,5
%A A137498 _Roger L. Bagula_, Apr 22 2008

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