OFFSET
1,1
COMMENTS
From Lorenzo Sauras Altuzarra, Oct 13 2020: (Start)
Conjecture I: if n > 2, then a(A005382(n))/12 is prime.
Conjecture II: if a(n)/12 is prime, then a(n-1)/12 - (n-1), a(n)/12 - n and a(n+2)/12 - (n+2) are multiples of 6. (End)
REFERENCES
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 257, Eq. 6.1.41.
L. V. Ahlfors, Complex Analysis, McGraw-Hill, 1979, p. 205
LINKS
Robert G. Wilson v, Table of n, a(n) for n = 1..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 257, Eq. 6.1.41.
Thomas Bayes, A letter to John Canton, Phil. Trans. Royal Society London, 53 (1763), 269-271.
R. P. Brent, Asymptotic approximation of central binomial coefficients with rigorous error bounds, arXiv:1608.04834 [math.NA], 2016.
N. Elezovic, Asymptotic Expansions of Central Binomial Coefficients and Catalan Numbers, J. Int. Seq. 17 (2014) # 14.2.1.
C. Impens, Stirling's series made easy, Am. Math. Monthly, 110 (No. 8, 2003), pp. 730-735.
Gergő Nemes, Generalization of Binet's Gamma function formulas, Integral Transforms and Special Functions, 24:8, pp. 597-606, 2013.
Eric Weisstein's World of Mathematics, Stirling's Series
FORMULA
From denominator of Jk(z) = (-1)^(k-1)*Bk/(((2k)*(2k-1))*z^(2k-1)), so Gamma(z) = sqrt(2pi)*z^(z-0.5)*exp(-z)*exp(J(z)).
MAPLE
a := n -> denom(bernoulli(2*n)/(2*n*(2*n-1))): # Lorenzo Sauras Altuzarra, Oct 13 2020
MATHEMATICA
Table[ Denominator[ BernoulliB[2n]/(2n(2n - 1))], {n, 31}] (* Robert G. Wilson v, Sep 21 2006 *)
s = LogGamma[z] + z - (z - 1/2) Log[z] - Log[2 Pi]/2 + O[z, Infinity]^62;
DeleteCases[CoefficientList[s, 1/z], 0] // Denominator (* Jean-François Alcover, Jun 13 2017 *)
PROG
(PARI) a(n)=if(n<1, 0, denominator(bernfrac(2*n)/(2*n)/(2*n-1)))
CROSSREFS
KEYWORD
frac,nonn,nice
AUTHOR
Douglas Stoll, dougstoll(AT)email.msn.com
EXTENSIONS
More terms from Frank Ellermann, Jun 13 2001
Bayes reference from Henry Bottomley, Jun 03 2003
STATUS
approved