A275994 Numerators of coefficients in the asymptotic expansion of the logarithm of the central binomial coefficient.

1, -1, 1, -17, 31, -691, 5461, -929569, 3202291, -221930581, 4722116521, -968383680827, 14717667114151, -2093660879252671, 86125672563201181, -129848163681107301953, 868320396104950823611, -209390615747646519456961, 14129659550745551130667441, -8486725345098385062639014237

Offset: 1

Author

Richard P. Brent, Sep 13 2016

Keywords

• frac
• sign

Comments

-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.

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

Examples

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

Links

• G. C. Greubel, Table of n, a(n) for n = 1..275 (terms 1..64 from Richard P. Brent)
• R. P. Brent, Asymptotic approximation of central binomial coefficients with rigorous error bounds, arXiv:1608.04834 [math.NA], 2016.

Crossrefs

Denominators are A275995.

Programs

Magma

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

Mathematica

Table[Numerator[(1 - 4^(-n)) BernoulliB[2 n] / (n (2 n - 1))], {n, 30}] (* Vincenzo Librandi, Sep 15 2016 *)

PARI

a(n) = numerator((1-4^(-n))*bernfrac(2*n)/(n*(2*n-1))); \\ Joerg Arndt, Sep 14 2016

Formula

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