A275995 Denominators of coefficients in the asymptotic expansion of the logarithm of the central binomial coefficient.

8, 192, 640, 14336, 18432, 180224, 425984, 15728640, 8912896, 79691776, 176160768, 3087007744, 3355443200, 28991029248, 62277025792, 4260607557632, 1133871366144, 9620726743040, 20340965113856, 343047627866112, 360639813910528, 3025855999639552, 6333186975989760, 211669182486413312

Offset: 1

Author

Richard P. Brent, Sep 13 2016

Keywords

• nonn
• frac

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 denominators of the coefficients t1, t2, t3, ... are given by this sequence.

The numerators are sequence A275994.

Examples

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

Links

• G. C. Greubel, Table of n, a(n) for n = 1..500 (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

Numerators are sequence A275994.

Programs

Magma

[Denominator((4^n-1)*BernoulliNumber(2*n)/4^n/n/(2*n-1)): n in [1..30]];

Mathematica

Table[Denominator[(1 - 4^(-n)) BernoulliB[2 n]/(n*(2*n - 1))], {n, 50}] (* G. C. Greubel, Feb 15 2017 *)

PARI

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

Formula

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