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A262382 Numerators of a semi-convergent series leading to the first Stieltjes constant gamma_1.

%I A262382 #32 Aug 28 2018 14:57:51
%S A262382 -1,11,-137,121,-7129,57844301,-1145993,4325053069,-1848652896341,
%T A262382 48069674759189,-1464950131199,105020512675255609,-22404210159235777,
%U A262382 1060366791013567384441,-15899753637685210768473787,2241672100026760127622163469,-8138835628210212414423299
%N A262382 Numerators of a semi-convergent series leading to the first Stieltjes constant gamma_1.
%C A262382 gamma_1 = - 1/12 + 11/720 - 137/15120 + 121/11200 - 7129/332640 + 57844301/908107200 - ..., see formulas (46)-(47) in the reference below.
%H A262382 G. C. Greubel, <a href="/A262382/b262382.txt">Table of n, a(n) for n = 1..259</a>
%H A262382 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 A262382 a(n) = numerator(-B_{2n}*H_{2n-1}/(2n)), where B_n and H_n are Bernoulli and harmonic numbers respectively.
%F A262382 a(n) = numerator(Zeta(1 - 2*n)*(Psi(2*n) + gamma)), where gamma is Euler's gamma. - _Peter Luschny_, Apr 19 2018
%e A262382 Numerators of -1/12, 11/720, -137/15120, 121/11200, -7129/332640, 57844301/908107200, ...
%p A262382 a := n -> numer(Zeta(1 - 2*n)*(Psi(2*n) + gamma)):
%p A262382 seq(a(n), n=1..16); # _Peter Luschny_, Apr 19 2018
%t A262382 a[n_] := Numerator[-BernoulliB[2*n]*HarmonicNumber[2*n - 1]/(2*n)]; Table[a[n], {n, 1, 20}]
%o A262382 (PARI) a(n) = numerator(-bernfrac(2*n)*sum(k=1,2*n-1,1/k)/(2*n)); \\ _Michel Marcus_, Sep 23 2015
%Y A262382 Cf. A001620, A002206, A195189, A075266, A262235, A001067, A006953, A082633, A262383 (denominators of this series), A086279, A086280, A262387.
%K A262382 frac,sign
%O A262382 1,2
%A A262382 _Iaroslav V. Blagouchine_, Sep 20 2015