%I #3 Mar 30 2012 17:34:26
%S 0,0,1,-3,6,6,-36,36,0,120,-360,240,-60,0,1800,-3600,1800,0,-2520,0,
%T 25200,-37800,15120,3360,0,-70560,0,352800,-423360,141120,0,241920,0,
%U -1693440,0,5080320,-5080320,1451520,-544320,0,10886400,0,-38102400,0,76204800,-65318400,16329600
%N 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].
%C Row sums: {0, 0, 1, 3, 6, 0, -60, 0, 3360, 0, -544320};
%F 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)).
%e {0},
%e {0},
%e {1},
%e {-3, 6},
%e {6, -36, 36},
%e {0, 120, -360, 240},
%e {-60, 0, 1800, -3600, 1800},
%e {0, -2520, 0, 25200, -37800, 15120},
%e {3360, 0, -70560, 0, 352800, -423360, 141120},
%e {0, 241920, 0, -1693440, 0, 5080320, -5080320,1451520},
%e {-544320, 0, 10886400, 0, -38102400, 0,76204800, -65318400, 16329600}
%t 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]
%K uned,tabl,sign
%O 1,4
%A _Roger L. Bagula_, Apr 22 2008