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A262387 Denominators of a semi-convergent series leading to the third Stieltjes constant gamma_3.

%I A262387 #20 Aug 28 2018 15:48:24
%S A262387 1,120,1008,28800,49896,101088000,5702400,12350257920000,
%T A262387 43480172736000,7075668600000,206069667148800,5919216795588096000,
%U A262387 581222138112000,8460252005694128640000,18991807088644406016000,1150594272774401495040000,33940540399314092544000,9737059611553100811150566400000,1290633707289706940160000,1263402804161736165764268432000000
%N A262387 Denominators of a semi-convergent series leading to the third Stieltjes constant gamma_3.
%C A262387 gamma_3 = + 1/120 - 17/1008 + 967/28800 - 4523/49896 + 33735311/101088000 - ..., see formulas (46)-(47) in the reference below.
%H A262387 G. C. Greubel, <a href="/A262387/b262387.txt">Table of n, a(n) for n = 1..500</a>
%H A262387 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>, 2015.
%F A262387 a(n) = denominator(-B_{2n}*(H^3_{2n-1}-3*H_{2n-1}*H^(2)_{2n-1}+2*H^(3)_{2n-1})/(2n)), where B_n, H_n and H^(k)_n are Bernoulli, harmonic and generalized harmonic numbers respectively.
%e A262387 Denominators of -0/1, 1/120, -17/1008, 967/28800, -4523/49896, 33735311/101088000, ...
%t A262387 a[n_] := Denominator[-BernoulliB[2*n]*(HarmonicNumber[2*n - 1]^3 - 3*HarmonicNumber[2*n - 1]*HarmonicNumber[2*n - 1, 2] + 2*HarmonicNumber[2*n - 1, 3])/(2*n)]; Table[a[n], {n, 1, 20}]
%o A262387 (PARI) a(n) = denominator(-bernfrac(2*n)*(sum(k=1,2*n-1,1/k)^3 -3*sum(k=1,2*n-1,1/k)*sum(k=1,2*n-1,1/k^2) + 2*sum(k=1,2*n-1,1/k^3))/(2*n));
%Y A262387 Cf. A001620, A002206, A195189, A075266, A262235, A001067, A006953, A082633, A262382, A262383, A086279, A262384, A262385, A086280, A262386 (numerators of this series).
%K A262387 nonn,frac
%O A262387 1,2
%A A262387 _Iaroslav V. Blagouchine_, Sep 20 2015