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

1,1

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

Numerators are given in A046968. Cf. A005382.

Sequence in context: A202926 A134800 A053068 * A074094 A012553 A128043

Adjacent sequences: A046966 A046967 A046968 * A046970 A046971 A046972

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