A262383 - OEIS

%I #27 Aug 29 2018 00:07:22

%S 12,720,15120,11200,332640,908107200,4324320,2940537600,175991175360,

%T 512143632000,1427794368,7795757249280,107084577600,279490747536000,

%U 200143324310529600,1178332991611776000,157531148611200,906996615309386784000,5828652498614400,262872227687509440000

%N Denominators of a semi-convergent series leading to the first Stieltjes constant gamma_1.

%C gamma_1 = - 1/12 + 11/720 - 137/15120 + 121/11200 - 7129/332640 + 57844301/908107200 - ..., see formulas (46)-(47) in the reference below.

%H G. C. Greubel, <a href="/A262383/b262383.txt">Table of n, a(n) for n = 1..500</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) = denominator(-B_{2n}*H_{2n-1}/(2n)), where B_n and H_n are Bernoulli and harmonic numbers respectively.

%F a(n) = denominator(Zeta(1 - 2*n)*(Psi(2*n) + gamma)), where gamma is Euler's gamma. - _Peter Luschny_, Apr 19 2018

%e Denominators of -1/12, 11/720, -137/15120, 121/11200, -7129/332640, 57844301/908107200, ...

%p a := n -> denom(Zeta(1 - 2*n)*(Psi(2*n) + gamma)):

%p seq(a(n), n=1..20); # _Peter Luschny_, Apr 19 2018

%t a[n_] := Denominator[-BernoulliB[2*n]*HarmonicNumber[2*n - 1]/(2*n)]; Table[a[n], {n, 1, 20}]

%o (PARI) a(n) = denominator(-bernfrac(2*n)*sum(k=1,2*n-1,1/k)/(2*n)); \\ _Michel Marcus_, Sep 23 2015

%Y Cf. A001620, A002206, A195189, A075266, A262235, A001067, A006953, A082633, A262382 (numerators of this series), A086279, A086280, A262387.

%K nonn,frac

%O 1,1

%A _Iaroslav V. Blagouchine_, Sep 20 2015