A122253 Binet's factorial series. Denominators of the coefficients of a convergent series for the logarithm of the Gamma function.
12, 12, 360, 60, 280, 168, 5040, 180, 11880, 264, 240240, 10920, 13104, 720, 367200, 3060, 813960, 15960, 1053360, 27720, 3825360, 16560, 5023200, 163800, 982800, 3024, 2630880, 6960, 33227040, 229152, 116867520, 235620, 282744, 2520, 1612119600, 7676760, 46060560
Offset: 1
Examples
c(1) = (1/1)*Integral_{x=0..1} x*(x - 1/2) dx = Integral_{x=0..1} (x^2 - x/2) dx = (x^3/3 - x^2/4)|{x=1} - (x^3/3 - x^2/4)|{x=0} = 1/12.
Links
- Raphael Schumacher, Rapidly Convergent Summation Formulas involving Stirling Series, arXiv preprint arXiv:1602.00336 [math.NT], 2016.
- Wikipedia, Stirling's approximation
Programs
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Maple
r := n -> add((-1)^(n-j)*Stirling1(n,j)*j/((j+1)*(j+2)), j = 1..n)/(2*n): a := n -> denom(r(n)); seq(a(n), n = 1..37); # Peter Luschny, Sep 22 2021
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Mathematica
Rising[z_, n_Integer/;n>0] := z Rising[z + 1, n - 1]; Rising[z_, 0] := 1; c[n_Integer/;n>0] := Integrate[Rising[x, n] (x - 1/2), {x, 0, 1}] / n; -
PARI
a(n) = denominator(sum(j=1, n, (-1)^(n-j)*stirling(n,j,1)*j/((j+1)*(j+2)))/(2*n)); \\ Michel Marcus, Sep 22 2021
Formula
c(n) = (1/n)*Integral_{x=0..1} x^n*(x - 1/2) dx.
Comments