Search: id:A137498
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%I A137498
%S A137498 0,0,0,0,6,60,120,300,1800,1800,0,12600,37800,25200,11760,0,352800,
%T A137498 705600,352800,0,846720,0,8467200,12700800,5080320,1814400,0,38102400,
               0,
%U A137498 190512000,228614400,76204800
%V A137498 0,0,0,0,6,-60,120,300,-1800,1800,0,12600,-37800,25200,-11760,0,352800,
               -705600,352800,
%W A137498 0,-846720,0,8467200,-12700800,5080320,1814400,0,-38102400,0,190512000,
               -228614400,
%X A137498 76204800
%N A137498 A triangular sequence of coefficients from a La Place 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:
%C A137498 {0, 0, 0, 0, 6, 60, 300, 0, -11760, 0, 1814400};
%C A137498 These functions are due 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]
%Y A137498 Sequence in context: A007358 A007357 A002827 this_sequence A036283 A126576 
               A121287
%Y A137498 Adjacent sequences: A137495 A137496 A137497 this_sequence A137499 A137500 
               A137501
%K A137498 uned,tabf,sign
%O A137498 1,5
%A A137498 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 22 2008

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