A075266 Numerator of the coefficient of x^n in log(-log(1-x)/x).
0, 1, 5, 1, 251, 19, 19087, 751, 1070017, 2857, 26842253, 434293, 703604254357, 8181904909, 1166309819657, 5044289, 8092989203533249, 5026792806787, 12600467236042756559, 69028763155644023, 8136836498467582599787
Offset: 0
Links
- Robert Israel, Table of n, a(n) for n = 1..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.
Programs
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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
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Mathematica
Numerator[ CoefficientList[ Series[ Log[ -Log[1 - x]/x], {x, 0, 20}], x]] -
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
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
Extensions
Edited by Robert G. Wilson v, Sep 17 2002
a(0) = 0 prepended by Peter Luschny, Aug 15 2025
Comments