%I #8 Dec 24 2018 21:40:43

%S 0,0,0,0,6,-60,120,300,-1800,1800,0,12600,-37800,25200,-11760,0,

%T 352800,-705600,352800,0,-846720,0,8467200,-12700800,5080320,1814400,

%U 0,-38102400,0,190512000,-228614400,76204800

%N 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 Row sums: {0, 0, 0, 0, 6, 60, 300, 0, -11760, 0, 1814400};

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

%F 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 {0},

%e {0},

%e {0},

%e {0},

%e {6},

%e {-60, 120},

%e {300, -1800, 1800},

%e {0, 12600, -37800, 25200},

%e {-11760, 0, 352800, -705600, 352800},

%e {0, -846720, 0, 8467200, -12700800, 5080320},

%e {1814400, 0, -38102400, 0, 190512000, -228614400, 76204800}

%t 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 uned,tabf,sign

%O 1,5

%A _Roger L. Bagula_, Apr 22 2008