This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A075266 #50 Aug 15 2025 11:05:06
%S A075266 0,1,5,1,251,19,19087,751,1070017,2857,26842253,434293,703604254357,
%T A075266 8181904909,1166309819657,5044289,8092989203533249,5026792806787,
%U A075266 12600467236042756559,69028763155644023,8136836498467582599787
%N A075266 Numerator of the coefficient of x^n in log(-log(1-x)/x).
%C A075266 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
%H A075266 Robert Israel, <a href="/A075266/b075266.txt">Table of n, a(n) for n = 1..447</a>
%H A075266 Iaroslav V. Blagouchine, <a href="http://dx.doi.org/10.1016/j.jmaa.2016.04.032">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</a>, Journal of Mathematical Analysis and Applications (Elsevier), 2016. <a href="http://arxiv.org/abs/1408.3902">arXiv version</a>, arXiv:1408.3902 [math.NT], 2014-2016
%H A075266 Iaroslav V. Blagouchine, <a href="http://dx.doi.org/10.1016/j.jnt.2015.06.012">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</a>, Journal of Number Theory (Elsevier), vol. 158, pp. 365-396, 2016. <a href="http://arxiv.org/abs/1501.00740">arXiv version</a>, arXiv:1501.00740 [math.NT], 2015.
%F A075266 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
%p A075266 S:= series(log(-log(1-x)/x),x,51):
%p A075266 seq(numer(coeff(S,x,j)),j=0..50); # _Robert Israel_, May 17 2016
%p A075266 # Alternative:
%p A075266 a := proc(n) local r; r := proc(n) option remember; if n=0 then 1 else
%p A075266 1 - add(r(k)/(n-k+1), k=0..n-1) fi end: numer(r(n)/(n*(n+1))) end:
%p A075266 seq(a(n), n=0..20); # _Peter Luschny_, Apr 19 2018
%t A075266 Numerator[ CoefficientList[ Series[ Log[ -Log[1 - x]/x], {x, 0, 20}], x]]
%o A075266 (SageMath)
%o A075266 @cached_function
%o A075266 def r(n): return 1 - sum(r(k)/(n-k+1) for k in range(n)) if n > 0 else 1
%o A075266 def a(n: int): return numerator(r(n)/(n*(n+1))) if n > 0 else 0
%o A075266 print([a(n) for n in range(21)]) # _Peter Luschny_, Aug 15 2025
%Y A075266 Cf. A053657, A075264, A075267 (denominator), A262235.
%K A075266 frac,nonn
%O A075266 0,3
%A A075266 _Paul D. Hanna_, Sep 15 2002
%E A075266 Edited by _Robert G. Wilson v_, Sep 17 2002
%E A075266 a(0) = 0 prepended by _Peter Luschny_, Aug 15 2025