A249646 - OEIS

%I #14 Sep 08 2022 08:46:10

%S 2,20,27720,5173168,2329089562800,2844937529085600,

%T 54749786241679275146400,1874681189225708508850515710400,

%U 718766754945489455304472257065075294400,153803387341307877636928566091115101174034840640

%N Denominators of 2*H(n)-H(n*(n+1)), a sequence the limit of which is gamma, the Euler-Mascheroni constant, where H(n) is the n-th harmonic number.

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.5 Euler-Mascheroni constant, p. 28.

%D J. J. Mačys, "A new problem," American Mathematical Monthly, (Jan 2012), vol. 119, no. 1, p. 82.

%H G. C. Greubel, <a href="/A249646/b249646.txt">Table of n, a(n) for n = 1..47</a>

%H StackExchange, <a href="http://math.stackexchange.com/questions/605354">Question on Macys formula for Euler-Mascheroni constant gamma</a>, Dec 13 2013.

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/Euler-MascheroniConstant.html">Euler-Mascheroni Constant</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant">Euler-Mascheroni constant</a>

%e Sequence of fractions begins 1/2, 11/20, 15619/27720, 2943155/5173168, 1331492839973/2329089562800, ...

%t Table[2*HarmonicNumber[n] - HarmonicNumber[n*(n + 1)] // Denominator, {n, 1, 10}]

%o (PARI) {a(n) = 2*sum(k=1,n, 1/k) - sum(k=1,n*(n+1), 1/k)};

%o for(n=1,15, print1(denominator(a(n)), ", ")) \\ _G. C. Greubel_, Sep 04 2018

%o (Magma) [Denominator(2*HarmonicNumber(n) - HarmonicNumber(n*(n + 1))): n in [1..15]]; // _G. C. Greubel_, Sep 04 2018

%Y Cf. A001008, A001620, A002805, A189048, A189049, A249645 (numerators).

%K nonn,frac

%O 1,1

%A _Jean-François Alcover_, Nov 03 2014