login
Denominators of expansion of (Sum_{k=1..n} 1/k) - log(n(1+1/(2n))) - gamma.
3

%I #16 Dec 26 2023 10:04:29

%S 24,24,960,160,8064,896,30720,4608,337920,22528,67092480,106496,

%T 688128,491520,267386880,2228224,1882718208,9961472,3460300800,

%U 44040192,6366953472,192937984,549621596160,838860800

%N Denominators of expansion of (Sum_{k=1..n} 1/k) - log(n(1+1/(2n))) - gamma.

%H E. Chlebus, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.118.03.268">A recursive scheme for improving the original rate of convergence to the Euler-Mascheroni constant</a>, Amer. Math. Mnthly, 118 (2011), 268-274.

%e 1/(24n^2) - 1/(24n^3) + 23/(960*n^4) - 1/(160n^5) - 11/(8064*n^6) - 1/(896n^7) + 143/(30720*n^8) + ...

%t s = Sum[1/k, {k, 1, n}] - Log[n*(1 + 1/(2*n))] - EulerGamma; CoefficientList[ Series[s, {n, Infinity, 25}], 1/n][[3 ;; -1]] // Denominator (* _Jean-François Alcover_, Sep 12 2013 *)

%Y Cf. A189048.

%K nonn,frac,more

%O 2,1

%A _N. J. A. Sloane_, Apr 16 2011

%E More terms from _Jean-François Alcover_, Sep 12 2013