OFFSET
0,3
COMMENTS
A series with these numerators leads to Euler's constant: gamma = 1 - 1/4 - 5/72 - 1/32 - 251/14400 - 19/1728 - 19087/2540160 - ..., see references [Blagouchine] below, as well as A262235. - Iaroslav V. Blagouchine, Sep 15 2015
LINKS
Robert Israel, Table of n, a(n) for n = 0..447
Iaroslav V. Blagouchine, Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to 1/pi, Journal of Mathematical Analysis and Applications (Elsevier), 2016. arXiv version, arXiv:1408.3902 [math.NT], 2014-2016
Iaroslav V. Blagouchine, Expansions of generalized Euler's constants into the series of polynomials in 1/pi^2 and into the formal enveloping series with rational coefficients only, Journal of Number Theory (Elsevier), vol. 158, pp. 365-396, 2016. arXiv version, arXiv:1501.00740 [math.NT], 2015.
FORMULA
a(n) = numerator(Sum_{k=1..n} (k-1)! * (-1)^(n-k-1) * binomial(n,k) * Stirling1(n+k,k) / (n+k)!). - Vladimir Kruchinin, Aug 14 2025
From Natalia L. Skirrow, Dec 05 2025: (Start)
n!*a(n)/A075267(n) is the derivative of the interpolating polynomial |Stirling1(n+x,x)| at x=0. (End)
MAPLE
S:= series(log(-log(1-x)/x), x, 51):
seq(numer(coeff(S, x, j)), j=0..50); # Robert Israel, May 17 2016
# Alternative:
a := proc(n) local r; r := proc(n) option remember; if n=0 then 1 else
1 - add(r(k)/(n-k+1), k=0..n-1) fi end: numer(r(n)/(n*(n+1))) end:
seq(a(n), n=0..20); # Peter Luschny, Apr 19 2018
MATHEMATICA
Numerator[ CoefficientList[ Series[ Log[ -Log[1 - x]/x], {x, 0, 20}], x]]
PROG
(SageMath)
@cached_function
def r(n): return 1 - sum(r(k)/(n-k+1) for k in range(n)) if n > 0 else 1
def a(n: int): return numerator(r(n)/(n*(n+1))) if n > 0 else 0
print([a(n) for n in range(21)]) # Peter Luschny, Aug 15 2025
CROSSREFS
KEYWORD
frac,nonn
AUTHOR
Paul D. Hanna, Sep 15 2002
EXTENSIONS
Edited by Robert G. Wilson v, Sep 17 2002
a(0) = 0 prepended by Peter Luschny, Aug 15 2025
STATUS
approved