A195189 Denominators of a sequence leading to gamma = A001620.

2, 24, 72, 2880, 800, 362880, 169344, 29030400, 9331200, 4790016000, 8673280, 31384184832000, 6181733376000, 439378587648000, 10346434560000, 512189896458240000, 265423814656, 14148260909088768000, 2076423318208512000, 96342919523794944000000, 74538995631567667200000

Offset: 0

Author

Paul Curtz, Sep 11 2011

Keywords

• nonn
• frac

Comments

gamma = 1/2 + 1/24 + 1/72 + 19/2880 + 3/800 + 863/362880 + 275/169344 + ... = (A002206 unsigned=reduced A141417(n+1)/A091137(n+1))/a(n) is an old formula based on Gregory's A002206/A002207.

This formula for Euler's constant was discovered circa 1780-1790 by the Italian mathematicians Gregorio Fontana (1735-1803) and Lorenzo Mascheroni (1750-1800), and was subsequently rediscovered several times (in particular, by Ernst Schröder in 1879, Niels E. Nørlund in 1923, Jan C. Kluyver in 1924, Charles Jordan in 1929, Kenter in 1999, and Victor Kowalenko in 2008). For more details, see references below. - Iaroslav V. Blagouchine, May 03 2015

Examples

			a(0)=1*2, a(1)=2*12, a(2)=3*24, a(3)=4*720.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..440
• Iaroslav V. Blagouchine, A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations, Journal of Number Theory (Elsevier), vol. 148, pp. 537-592 and vol. 151, pp. 276-277, 2015. arXiv version.
• 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.
• M. Coffey and J. Sondow, Rebuttal of Kowalenko's paper as concerns the irrationality of Euler's constant, Acta Appl. Math., 121 (2012), 1-3.
• J. C. Kluyver, Euler's constant and natural numbers, Proc. Kon. Ned. Akad. Wet., 27(1-2) (1924), 142-144.

Crossrefs

Cf. A001620, A002206, A002207, A091137, A141417.

Programs

Mathematica

g[n_]:=Sum[StirlingS1[n,l]/(l+1),{l,1,n}]/(n*n!); a[n_]:=Denominator[g[n]]; Table[a[n],{n,1,30}] (* Iaroslav V. Blagouchine, May 03 2015 *)
g[n_] := Sum[ BernoulliB[j]/j * StirlingS1[n, j-1], {j, 1, n+1}] / n! ; a[n_] := (n+1)*Denominator[g[n]]; Table[a[n], {n, 0, 20}]
(* or *) max = 20; Denominator[ CoefficientList[ Series[ 1/Log[1 + x] - 1/x, {x, 0, max}], x]]*Range[max+1] (* Jean-François Alcover, Sep 04 2013 *)

Formula

a(n) = (n+1) * A002207(n).

Extensions

More terms from Jean-François Alcover, Sep 04 2013