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A275994
Numerators of coefficients in the asymptotic expansion of the logarithm of the central binomial coefficient
3
1, -1, 1, -17, 31, -691, 5461, -929569, 3202291, -221930581, 4722116521, -968383680827, 14717667114151, -2093660879252671, 86125672563201181, -129848163681107301953, 868320396104950823611, -209390615747646519456961, 14129659550745551130667441, -8486725345098385062639014237
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)
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 A335360
KEYWORD
frac,sign
AUTHOR
Richard P. Brent, Sep 13 2016
STATUS
approved