proposed

approved

editing

proposed

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

{0, 0, 0, 0, 6, 60, 300, 0, -11760, 0, 1814400};

proposed

editing

editing

proposed

A triangular sequence of coefficients from a La Place 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].

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

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)).

approved

editing

_Roger L. Bagula (rlbagulatftn(AT)yahoo.com), _, Apr 22 2008

nonn,uned,tabf,newsign

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].

0, 0, 0, 0, 6, -60, 120, 300, -1800, 1800, 0, 12600, -37800, 25200, -11760, 0, 352800, -705600, 352800, 0, -846720, 0, 8467200, -12700800, 5080320, 1814400, 0, -38102400, 0, 190512000, -228614400, 76204800

1,5

Row sums:

{0, 0, 0, 0, 6, 60, 300, 0, -11760, 0, 1814400};

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

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)).

{0},

{0},

{0},

{0},

{6},

{-60, 120},

{300, -1800, 1800},

{0, 12600, -37800, 25200},

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

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

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

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]

nonn,uned,tabf

Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 22 2008

approved