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A275994 Numerators of coefficients in the asymptotic expansion of the logarithm of the central binomial coefficient 2
1, -1, 1, -17, 31, -691, 5461, -929569, 3202291, -221930581, 4722116521, -968383680827, 14717667114151, -2093660879252671, 86125672563201181, -129848163681107301953, 868320396104950823611, -209390615747646519456961, 14129659550745551130667441, -8486725345098385062639014237 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

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).

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.

FORMULA

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

EXAMPLE

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

MATHEMATICA

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

PROG

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

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

CROSSREFS

Denominators are A275995.

Sequence in context: A279370 A276592 A002425 * A046990 A059212 A058899

Adjacent sequences:  A275991 A275992 A275993 * A275995 A275996 A275997

KEYWORD

frac,sign

AUTHOR

Richard P. Brent, Sep 13 2016

STATUS

approved

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Last modified October 20 22:38 EDT 2018. Contains 316404 sequences. (Running on oeis4.)