cp's OEIS Frontend

A122253 Binet's factorial series. Denominators of the coefficients of a convergent series for the logarithm of the Gamma function.

%I A122253 #29 Sep 22 2021 10:29:08
%S A122253 12,12,360,60,280,168,5040,180,11880,264,240240,10920,13104,720,
%T A122253 367200,3060,813960,15960,1053360,27720,3825360,16560,5023200,163800,
%U A122253 982800,3024,2630880,6960,33227040,229152,116867520,235620,282744,2520,1612119600,7676760,46060560
%N A122253 Binet's factorial series. Denominators of the coefficients of a convergent series for the logarithm of the Gamma function.
%C A122253 See A122252 for references and formulas.
%H A122253 Raphael Schumacher, <a href="http://arxiv.org/abs/1602.00336">Rapidly Convergent Summation Formulas involving Stirling Series</a>, arXiv preprint arXiv:1602.00336 [math.NT], 2016.
%H A122253 Wikipedia, <a href="http://en.wikipedia.org/wiki/Stirling%27s_approximation">Stirling's approximation</a>
%F A122253 c(n) = (1/n)*Integral_{x=0..1} x^n*(x - 1/2) dx.
%e A122253 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.
%p A122253 r := n -> add((-1)^(n-j)*Stirling1(n,j)*j/((j+1)*(j+2)), j = 1..n)/(2*n):
%p A122253 a := n -> denom(r(n)); seq(a(n), n = 1..37); # _Peter Luschny_, Sep 22 2021
%t A122253 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;
%o A122253 (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
%Y A122253 Cf. A122252 (numerators), A001163, A001164.
%K A122253 easy,frac,nonn
%O A122253 1,1
%A A122253 Paul Drees (zemyla(AT)gmail.com), Aug 27 2006