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Binet's factorial series. Numerators of the coefficients of a convergent series for the logarithm of the Gamma function.
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%I #32 Sep 22 2021 10:29:04

%S 1,1,59,29,533,1577,280361,69311,36226519,7178335,64766889203,

%T 32128227179,459253205417,325788932161,2311165698322609,

%U 287144996287039,1215091897184850539,402833263943353393,476099430416027805187,236881416523193720213,650730651653461090091101

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

%H Robert G. Wilson v, <a href="/A122252/b122252.txt">Table of n, a(n) for n = 1..100</a>

%H 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 Ch. Hermite, <a href="http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002163330">Sur la function log Gamma(a)</a> Journal für die reine und angewandte Mathematik, 115:201-208, 1895.

%H 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 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 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 Wikipedia, <a href="http://en.wikipedia.org/wiki/Stirling%27s_approximation">Stirling's Approximation</a>

%F a(n) = numerator(c(n)), where c(n) are given by Binet's formulas:

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

%F 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 Rational sequence starts: 1/12, 1/12, 59/360, 29/60, 533/280, 1577/168, 280361/5040, ...

%e 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 r := n -> add((-1)^(n-j)*Stirling1(n,j)*j/((j+1)*(j+2)), j=1..n)/(2*n):

%p a := n -> numer(r(n)); seq(a(n), n=1..21); # _Peter Luschny_, Sep 22 2021

%t 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 (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 Cf. A122253 (denominators), A001163, A001164.

%K easy,frac,nonn

%O 1,3

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

%E Edited by _Peter Luschny_, Sep 22 2021