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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].
0

%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