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A122252 Binet's factorial series. Numerators of the coefficients of a convergent series for the logarithm of the Gamma function. 2
1, 1, 59, 29, 533, 1577, 280361, 69311, 36226519, 7178335, 64766889203, 32128227179, 459253205417, 325788932161, 2311165698322609, 287144996287039, 1215091897184850539, 402833263943353393, 476099430416027805187, 236881416523193720213, 650730651653461090091101 (list; graph; refs; listen; history; text; internal format)
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.

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).

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

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

CROSSREFS

Cf. A122253 (denominators), A001163, A001164.

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

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Last modified January 17 20:06 EST 2022. Contains 350410 sequences. (Running on oeis4.)