|
%I #35 Apr 20 2018 01:42:10
%S 1,5,1,251,19,19087,751,1070017,2857,26842253,434293,703604254357,
%T 8181904909,1166309819657,5044289,8092989203533249,5026792806787,
%U 12600467236042756559,69028763155644023,8136836498467582599787
%N Numerator of the coefficient of x^n in log(-log(1-x)/x).
%C 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 Robert Israel, <a href="/A075266/b075266.txt">Table of n, a(n) for n = 1..447</a>
%H 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 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.
%p S:= series(log(-log(1-x)/x),x,51):
%p seq(numer(coeff(S,x,j)),j=1..50); # _Robert Israel_, May 17 2016
%p # Alternative:
%p a := proc(n) local r; r := proc(n) option remember; if n=0 then 1 else
%p 1 - add(r(k)/(n-k+1), k=0..n-1) fi end: numer(r(n)/(n*(n+1))) end:
%p seq(a(n), n=1..20); # _Peter Luschny_, Apr 19 2018
%t Numerator[ CoefficientList[ Series[ Log[ -Log[1 - x]/x], {x, 0, 20}], x]]
%Y Cf. A053657, A075264, A075267, A262235.
%K frac,nonn
%O 1,2
%A _Paul D. Hanna_, Sep 15 2002
%E Edited by _Robert G. Wilson v_, Sep 17 2002
|
