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%I #37 Aug 29 2018 02:50:54
%S 0,-1,5,-469,6515,-131672123,63427,-47800416479,15112153995391,
%T -29632323552377537,4843119962464267,-1882558877249847563479,
%U 2432942522372150087,-2768809380553055597986831,334463513629004852735064113,-1125061940756859461946444233539,333807583501528759350875247323
%N Numerators of a semi-convergent series leading to the second Stieltjes constant gamma_2.
%C gamma_2 = - 1/60 + 5/336 - 469/21600 + 6515/133056 - 131672123/825552000 + ..., see formulas (46)-(47) in the reference below.
%H G. C. Greubel, <a href="/A262384/b262384.txt">Table of n, a(n) for n = 1..237</a>
%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.
%F a(n) = numerator(B_{2n}*(H^2_{2n-1}-H^(2)_{2n-1})/(2n)), where B_n, H_n and H^(k)_n are Bernoulli, harmonic and generalized harmonic numbers respectively.
%F a(n) = numerator(-Zeta(1 - 2*n)*(Psi(1,2*n) + (Psi(0,2*n) + gamma)^2 - (Pi^2)/6)), where gamma is Euler's gamma and Psi is the digamma function. - _Peter Luschny_, Apr 19 2018
%e Numerators of 0/1, -1/60, 5/336, -469/21600, 6515/133056, -131672123/825552000, ...
%p a := n -> numer(-Zeta(1 - 2*n)*(Psi(1, 2*n) + (Psi(0,2*n) + gamma)^2 - (Pi^2)/6)):
%p seq(a(n), n=1..17); # _Peter Luschny_, Apr 19 2018
%t a[n_] := Numerator[BernoulliB[2*n]*(HarmonicNumber[2*n - 1]^2 - HarmonicNumber[2*n - 1, 2])/(2*n)]; Table[a[n], {n, 1, 20}]
%o (PARI) a(n) = numerator(bernfrac(2*n)*(sum(k=1,2*n-1,1/k)^2 - sum(k=1,2*n-1,1/k^2))/(2*n)); \\ _Michel Marcus_, Sep 23 2015
%Y The sequence of denominators is A262385.
%Y Cf. A001067, A001620, A002206, A006953, A075266, A082633, A086279, A086280, A195189, A262235, A262382, A262383, A262386, A262387.
%K frac,sign
%O 1,3
%A _Iaroslav V. Blagouchine_, Sep 20 2015
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