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

1,1

Comments

See A122252 for references and formulas.

Links

Table of n, a(n) for n=1..37.

Raphael Schumacher, Rapidly Convergent Summation Formulas involving Stirling Series, arXiv preprint arXiv:1602.00336 [math.NT], 2016.

Wikipedia, Stirling's approximation

Formula

c(n) = (1/n)*Integral_{x=0..1} x^n*(x - 1/2) dx.

Example

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.

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;

Prog

(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

Crossrefs

Cf. A122252 (numerators), A001163, A001164.

Sequence in context: A038338 A221796 A222298 * A156456 A077180 A105745

Adjacent sequences: A122250 A122251 A122252 * A122254 A122255 A122256

Keyword

easy,frac,nonn

Author

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

Status

approved