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A275994
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Numerators of coefficients in the asymptotic expansion of the logarithm of the central binomial coefficient
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3
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1, -1, 1, -17, 31, -691, 5461, -929569, 3202291, -221930581, 4722116521, -968383680827, 14717667114151, -2093660879252671, 86125672563201181, -129848163681107301953, 868320396104950823611, -209390615747646519456961, 14129659550745551130667441, -8486725345098385062639014237
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OFFSET
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1,4
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COMMENTS
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-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).
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LINKS
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FORMULA
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a(n) = numerator((1-4^(-n))*Bernoulli(2*n)/(n*(2*n-1))).
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EXAMPLE
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For n = 4, a(4) = numerator(-17/13336) = -17.
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MATHEMATICA
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Table[Numerator[(1 - 4^(-n)) BernoulliB[2 n] / (n (2 n - 1))], {n, 30}] (* Vincenzo Librandi, Sep 15 2016 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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frac,sign
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AUTHOR
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STATUS
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approved
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