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

Author

Paul Drees (zemyla(AT)gmail.com), Aug 27 2006

Keywords

• easy
• frac
• nonn

Comments

See A122252 for references and formulas.

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

Crossrefs

Cf. A122252 (numerators), A001163, A001164.

Programs

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

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.