

A195189


Denominators of a sequence leading to gamma = A001620.


14



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

0,1


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 17801790 by the Italian mathematicians Gregorio Fontana (17351803) and Lorenzo Mascheroni (17501800), 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


LINKS

G. C. Greubel, Table of n, a(n) for n = 0..440
Iaroslav V. Blagouchine, A theorem for the closedform evaluation of the first generalized Stieltjes constant at rational arguments and some related summations, Journal of Number Theory (Elsevier), vol. 148, pp. 537592 and vol. 151, pp. 276277, 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. 365396, 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), 13.
J. C. Kluyver, Euler's constant and natural numbers, Proc. Kon. Ned. Akad. Wet., 27(12) (1924), 142144.


FORMULA

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


EXAMPLE

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


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, j1], {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] (* JeanFrançois Alcover, Sep 04 2013 *)


CROSSREFS

Cf. A001620, A002206, A002207, A091137, A141417.
Sequence in context: A152965 A139284 A003614 * A119060 A119050 A119068
Adjacent sequences: A195186 A195187 A195188 * A195190 A195191 A195192


KEYWORD

nonn


AUTHOR

Paul Curtz, Sep 11 2011


EXTENSIONS

More terms from JeanFrançois Alcover, Sep 04 2013


STATUS

approved



