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A122252 Binet's factorial series. Numerators of the coefficients of a convergent series for the logarithm of the Gamma function.

%I A122252 #34 Aug 11 2025 07:23:37
%S A122252 1,1,59,29,533,1577,280361,69311,36226519,7178335,64766889203,
%T A122252 32128227179,459253205417,325788932161,2311165698322609,
%U A122252 287144996287039,1215091897184850539,402833263943353393,476099430416027805187,236881416523193720213,650730651653461090091101
%N A122252 Binet's factorial series. Numerators of the coefficients of a convergent series for the logarithm of the Gamma function.
%H A122252 Robert G. Wilson v, <a href="/A122252/b122252.txt">Table of n, a(n) for n = 1..100</a>
%H A122252 J. P. M. Binet, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k65355594.texteImage">Mémoire sur les intégrales définites Eulériennes et sur leur application à la théorie des suites ainsi qu'à l`évaluation des functions des grands nombres</a>, Journal de l`École Polytechnique, XVI:123-343, July 1839.
%H A122252 Ch. Hermite, <a href="https://gdz.sub.uni-goettingen.de/id/PPN243919689_0115">Sur la function log Gamma(a)</a> Journal für die reine und angewandte Mathematik, 115:201-208, 1895.
%H A122252 G. Nemes, <a href="https://doi.org/10.1080/10652469.2012.725168">Generalization of Binet's Gamma function formulas</a>, Integral Transforms and Special Functions, 24(8):595-606, 2013.
%H A122252 Raphael Schumacher, <a href="http://arxiv.org/abs/1602.00336">Rapidly Convergent Summation Formulas involving Stirling Series</a>, arXiv:1602.00336 [math.NT], 2016.
%H A122252 P. Van Mieghem, <a href="https://arxiv.org/abs/2102.04891">Binet's factorial series and extensions to Laplace transforms</a>, arXiv:2102.04891 [math.FA], 2021.
%H A122252 Wikipedia, <a href="http://en.wikipedia.org/wiki/Stirling%27s_approximation">Stirling's Approximation</a>
%F A122252 a(n) = numerator(c(n)), where c(n) are given by Binet's formulas:
%F A122252 log Gamma z = (z - 1/2) log z - z + log(2*Pi)/2 + Sum_{n >= 1} c(n)/(z+1)^(n), where z^(n) is the rising factorial.
%F A122252 c(n) = (1/n)*Integral_{x=0..1} x^(n)*(x - 1/2).
%F A122252 a(n) = numerator((1/2n)*Sum_{j=1..n} (-1)^(n-j)*Stirling1(n,j)*j/((j+1)*(j+2))). - _Peter Luschny_, Sep 22 2021
%e A122252 Rational sequence starts: 1/12, 1/12, 59/360, 29/60, 533/280, 1577/168, 280361/5040, ...
%e A122252 c(1) = Integral_{x=0..1} x*(x - 1/2) / 1 = Integral_{x=0..1} (x^2 - x/2) = (x^3/3 - x^2/4) | {x, 0, 1} = 1/12.
%p A122252 r := n -> add((-1)^(n-j)*Stirling1(n,j)*j/((j+1)*(j+2)), j=1..n)/(2*n):
%p A122252 a := n -> numer(r(n)); seq(a(n), n=1..21); # _Peter Luschny_, Sep 22 2021
%t A122252 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; Numerator@ Array[c, 19] (* updated by _Robert G. Wilson v_, Aug 15 2015 *)
%o A122252 (PARI) a(n) = numerator(sum(j=1, n, (-1)^(n-j)*stirling(n,j,1)*j/((j+1)*(j+2)))/(2*n)); \\ _Michel Marcus_, Sep 22 2021
%Y A122252 Cf. A122253 (denominators), A001163, A001164.
%K A122252 easy,frac,nonn
%O A122252 1,3
%A A122252 Paul Drees (zemyla(AT)gmail.com), Aug 27 2006
%E A122252 Edited by _Peter Luschny_, Sep 22 2021