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

1, 1, 59, 29, 533, 1577, 280361, 69311, 36226519, 7178335, 64766889203, 32128227179, 459253205417, 325788932161, 2311165698322609, 287144996287039, 1215091897184850539, 402833263943353393, 476099430416027805187, 236881416523193720213, 650730651653461090091101

Offset

1,3

Links

Robert G. Wilson v, Table of n, a(n) for n = 1..100

J. P. M. Binet, 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, Journal de l`École Polytechnique, XVI:123-343, July 1839.

Ch. Hermite, Sur la function log Gamma(a) Journal für die reine und angewandte Mathematik, 115:201-208, 1895.

G. Nemes, Generalization of Binet's Gamma function formulas, Integral Transforms and Special Functions, 24(8):595-606, 2013.

Peter O., Is there a tighter error bound for this convergent version of Stirling's series?, Mathematics StackExchange, 2020.

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

P. Van Mieghem, Binet's factorial series and extensions to Laplace transforms, arXiv:2102.04891 [math.FA], 2021.

Wikipedia, Stirling's Approximation

Formula

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

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.

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

c(n) = (1/2n)*Sum_{j=1..n} (-1)^(n-j)*Stirling1(n,j)*j/((j+1)*(j+2)). - Peter Luschny, Sep 22 2021

From Natalia L. Skirrow, Apr 30 2026: (Start)

Let G(n) = A002206(n-1)/A002207(n-1) be the Gregory coefficients and H(n) = (-1)^n*G(n). (H(0)=1, H(n) = -|G(n)| for n>0.)

c(n) = (n-1)!*( (n+1)*H(n+1) + Sum_{j=0..n} H(j)/2 ).

c(n) = (n-1)!*( (n+1)*H(n+1) - Sum_{j>=n+2} H(j) )/2.

c(n) ~ (n-1)!/(2*log(n)).

E.g.f. for c(n+1): ( 1/x + 1/(1-x)*(1/2 - 1/log(1/(1-x))) ) / log(1/(1-x)).

(End)

Example

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

Maple

r := n -> add((-1)^(n-j)*Stirling1(n, j)*j/((j+1)*(j+2)), j=1..n)/(2*n):
a := n -> numer(r(n)); seq(a(n), n=1..21); # 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; Numerator@ Array[c, 19] (* updated by Robert G. Wilson v, Aug 15 2015 *)

Prog

(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
(Python)
from math import factorial as fact
from functools import lru_cache
from fractions import Fraction as frac
gregory=lru_cache(lambda n: sum(frac((-1)**k, k)*gregory(n+1-k) for k in range(2, n+2)) if n else 1)
c=lambda n: fact(n-1)*((n+1)*(-1)**(n-1)*gregory(n+1)+sum((-1)**j*gregory(j) for j in range(n+1))/2)
A122252=lambda n: c(n).numerator # Natalia L. Skirrow, Apr 30 2026

Crossrefs

Cf. A122253 (denominators), A001163, A001164, A002206/A002207 (Gregory coefficients).

Sequence in context: A145532 A152214 A033379 * A119945 A054379 A278372

Adjacent sequences: A122249 A122250 A122251 * A122253 A122254 A122255

Keyword

easy,frac,nonn

Author

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

Extensions

Edited by Peter Luschny, Sep 22 2021

Status

approved