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

1, 43, 20431, 2150797323119, 9020112358835722225404403, 51551916515442115079024221439308876243677598340510141

Offset: 1

Author

Iaroslav V. Blagouchine, Oct 03 2015

Keywords

• frac
• nonn

Comments

gamma = 1 - 1/12 - 43/420 - 20431/240240 - 2150797323119/36100888223400 - ..., see formula (36) in the reference below.

Examples

			Numerators of 1/12, 43/420, 20431/240240, 2150797323119/36100888223400, ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10
• Iaroslav V. Blagouchine, 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. Journal of Number Theory (Elsevier), vol. 158, pp. 365-396, 2016. arXiv version, arXiv:1501.00740 [math.NT], 2015.

Crossrefs

Cf. A075266, A075267, A001620, A195189, A002657, A002790, A262235, A075266, A006953, A001067, A262858 (denominators of this series).

Programs

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

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

Mathematica

a[n_] := Numerator[n*Sum[(-1)^(k + 1)/k, {k, 2^n + 1, 2^(n + 1)}]]; Table[a[n], {n, 1, 8}]

PARI

a(n) = numerator(n*sum(k=2^n + 1,2^(n + 1),(-1)^(k + 1)/k));

Formula

a(n) = n * Sum_{k = 2^n + 1 .. 2^(n + 1)} (-1)^(k + 1)/k.