A046969 Denominators of coefficients in Stirling's expansion for log(Gamma(z)).

12, 360, 1260, 1680, 1188, 360360, 156, 122400, 244188, 125400, 5796, 1506960, 300, 93960, 2492028, 505920, 396, 2418179400, 444, 21106800, 3109932, 118680, 25380, 104700960, 6468, 324360, 2283876, 382800, 40356, 201025024200, 732

Offset: 1

Author

Douglas Stoll, dougstoll(AT)email.msn.com

Keywords

• frac
• nonn
• nice

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

Crossrefs

Numerators are given in A046968. Cf. A005382.

Programs

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 *)

PARI

a(n)=if(n<1,0,denominator(bernfrac(2*n)/(2*n)/(2*n-1)))

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)).

Extensions

More terms from Frank Ellermann, Jun 13 2001

Bayes reference from Henry Bottomley, Jun 03 2003