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A262856 Numerators of the Nielsen-Jacobsthal series leading to Euler's constant.

%I A262856 #21 Sep 08 2022 08:46:14
%S A262856 1,43,20431,2150797323119,9020112358835722225404403,
%T A262856 51551916515442115079024221439308876243677598340510141
%N A262856 Numerators of the Nielsen-Jacobsthal series leading to Euler's constant.
%C A262856 gamma = 1 - 1/12 - 43/420 - 20431/240240 - 2150797323119/36100888223400 - ..., see formula (36) in the reference below.
%H A262856 G. C. Greubel, <a href="/A262856/b262856.txt">Table of n, a(n) for n = 1..10</a>
%H A262856 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 A262856 a(n) = n * Sum_{k = 2^n + 1 .. 2^(n + 1)} (-1)^(k + 1)/k.
%e A262856 Numerators of 1/12, 43/420, 20431/240240, 2150797323119/36100888223400, ...
%t A262856 a[n_] := Numerator[n*Sum[(-1)^(k + 1)/k, {k, 2^n + 1, 2^(n + 1)}]]; Table[a[n], {n, 1, 8}]
%o A262856 (PARI) a(n) = numerator(n*sum(k=2^n + 1,2^(n + 1),(-1)^(k + 1)/k));
%o A262856 (Magma) [Numerator(n*(&+[(-1)^(k+1)/k: k in [2^n+1..2^(n+1)]])): n in [1..6]]; // _G. C. Greubel_, Oct 28 2018
%o A262856 (GAP) List(List([1..6],n->n*Sum([2^n+1..2^(n+1)],k->(-1)^(k+1)/k)),NumeratorRat); # _Muniru A Asiru_, Oct 29 2018
%Y A262856 Cf. A075266, A075267, A001620, A195189, A002657, A002790, A262235, A075266, A006953, A001067, A262858 (denominators of this series).
%K A262856 frac,nonn
%O A262856 1,2
%A A262856 _Iaroslav V. Blagouchine_, Oct 03 2015