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

1,1

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.

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.

Formula

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

Example

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

Mathematica

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

Prog

(Magma) [Denominator((4^n-1)*BernoulliNumber(2*n)/4^n/n/(2*n-1)): n in [1..30]];
(PARI) a(n) = denominator((1-4^(-n))*bernfrac(2*n)/(n*(2*n-1))); \\ Joerg Arndt, Sep 14 2016

Crossrefs

Numerators are sequence A275994.

Sequence in context: A366150 A251669 A158654 * A129004 A267948 A339487

Adjacent sequences: A275992 A275993 A275994 * A275996 A275997 A275998

Keyword

nonn,frac

Author

Richard P. Brent, Sep 13 2016

Status

approved