%I #8 Dec 24 2018 21:41:04
%S 0,0,0,0,0,24,-360,720,2520,-15120,15120,0,141120,-423360,282240,
%T -169344,0,5080320,-10160640,5080320,0,-15240960,0,152409600,
%U -228614400,91445760
%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[6,1+1/t-x].
%C Row sums: {0, 0, 0, 0, 0, 24, 360, 2520, 0, -169344, 0};
%C These functions are due to the close connection of the Bernoulli-type functions with the Zeta (generalized) functions.
%F Zeta[6,1+1/t-x] = Sum[1/(n+1/t+x)^6,{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 {0},
%e {24},
%e {-360,720},
%e {2520, -15120, 15120},
%e {0, 141120, -423360, 282240},
%e {-169344, 0, 5080320, -10160640, 5080320},
%e {0, -15240960, 0, 152409600, -228614400, 91445760}
%t LaplaceTransform[t*Exp[x*t]/(Exp[t] - 1), t, s]; Clear[p, f, g] p[t_] = Zeta[6, 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 1,6
%A _Roger L. Bagula_, Apr 22 2008