A275994 - OEIS

%I #32 Sep 08 2022 08:46:17

%S 1,-1,1,-17,31,-691,5461,-929569,3202291,-221930581,4722116521,

%T -968383680827,14717667114151,-2093660879252671,86125672563201181,

%U -129848163681107301953,868320396104950823611,-209390615747646519456961,14129659550745551130667441,-8486725345098385062639014237

%N Numerators of coefficients in the asymptotic expansion of the logarithm of the central binomial coefficient

%C -log(binomial(2n,n)) + log(4^n/sqrt(Pi*n)) has an asymptotic expansion (t1/n + t2/n^3 + t3/n^5 + ...) where the numerators of the coefficients t1, t2, t3, ... are given by this sequence.

%C The sequence is different from A002425, but the first difference is at index 60 (see the text files).

%H G. C. Greubel, <a href="/A275994/b275994.txt">Table of n, a(n) for n = 1..275</a> (terms 1..64 from Richard P. Brent)

%H R. P. Brent, <a href="http://arxiv.org/abs/1608.04834"> Asymptotic approximation of central binomial coefficients with rigorous error bounds</a>, arXiv:1608.04834 [math.NA], 2016.

%F a(n) = numerator((1-4^(-n))*Bernoulli(2*n)/(n*(2*n-1))).

%e For n = 4, a(4) = numerator(-17/13336) = -17.

%t Table[Numerator[(1 - 4^(-n)) BernoulliB[2 n] / (n (2 n - 1))], {n, 30}] (* _Vincenzo Librandi_, Sep 15 2016 *)

%o (Magma) [Numerator((4^n-1)*BernoulliNumber(2*n)/4^n/n/(2*n-1)): n in [1..20]];

%o (PARI) a(n) = numerator((1-4^(-n))*bernfrac(2*n)/(n*(2*n-1))); \\ _Joerg Arndt_, Sep 14 2016

%Y Denominators are A275995.

%K frac,sign

%O 1,4

%A _Richard P. Brent_, Sep 13 2016