A377024 Decimal expansion of the constant F(2) related to asymptotic products of factorials.

1, 0, 2, 3, 9, 3, 7, 4, 1, 1, 6, 3, 7, 1, 1, 8, 4, 0, 1, 5, 7, 7, 9, 5, 0, 7, 8, 2, 5, 8, 6, 2, 1, 7, 8, 0, 0, 8, 0, 3, 7, 6, 0, 9, 8, 0, 4, 3, 6, 4, 4, 0, 0, 5, 1, 2, 9, 4, 6, 9, 9, 0, 9, 5, 1, 3, 4, 7, 6, 9, 2, 4, 1, 2, 4, 0, 0, 7, 8, 2, 7, 6, 8, 7, 1, 1, 5, 2, 9, 4, 7, 4, 6, 5, 9, 8, 8, 1, 7, 3, 0, 6, 2, 3, 4, 8, 3, 6, 4, 2, 4

Offset: 1

Author

Bernd C. Kellner, Oct 13 2024

Keywords

• nonn
• cons

Comments

The constants F(1) = A213080, F(2), ... occur in the context of asymptotic constants related to asymptotic products of factorials as well as of binomial and multinomial coefficients. Moreover, the sequence (F(k)){k >= 1} is strictly decreasing with limit 1. For example, for k >= 1 the asymptotic product Prod{v >= 1} (k*v)! has the asymptotic constant F(k)*A^k*(2*Pi)^(1/4), where A = A074962 denotes the Glaisher-Kinkelin constant. Let gamma = A001620 be Euler's constant and Gamma(x) be the gamma function.

For k >= 1, the constants F(k) can be computed by an explicit formula and a divergent series expansion, as follows. We have log(F(k)) = (1/(12*k))*(1-log(k)) + (k/4)*log(2*Pi) - ((k^2+1)/k)*log(A) - Sum_{v=1..k-1} (v/k)*log(Gamma(v/k)) = gamma/(12*k) - t*zeta(3)/(360*k^3) with some t in (0,1), respectively.

It follows that log(F(2)) = 1/24 + log(2*Pi)/4 + (5/24)*log(2) - (5/2)*log(A) = gamma/24 - t*zeta(3)/2880 with some t in (0,1), and so F(2) lies in the interval (1.023914..., 1.024342...) (see Kellner 2009 and 2024).

Examples

			1.02393741163711840157795078258621780080376098043644005129469909513476924124007...
		

Links

• Bernd C. Kellner, Table of n, a(n) for n = 1..10000
• Bernd C. Kellner, On asymptotic constants related to products of Bernoulli numbers and factorials, Integers 9 (2009), Article #A08, 83-106; alternative link; arXiv:0604505 [math.NT], 2006.
• Bernd C. Kellner, Asymptotic products of binomial and multinomial coefficients revisited, Integers 24 (2024), Article #A59, 10 pp.; arXiv:2312.11369 [math.CO], 2023.

Crossrefs

Cf. A001620, A002117, A074962, A213080, A377023.

Programs

Maple

exp(-1/6+5/2*Zeta(1, -1))*(2*Pi)^(1/4)*2^(5/24); evalf(%, 100);

Mathematica

RealDigits[Exp[1/24] (2 Pi)^(1/4) 2^(5/24) / Glaisher^(5/2), 10, 100][[1]]

PARI

default(realprecision, 100);
exp(-1/6+5/2*zeta'(-1))*(2*Pi)^(1/4)*2^(5/24)

Sage

import mpmath
mpmath.mp.pretty = True; mpmath.mp.dps = 100
mpmath.exp(-1/6+5/2*mpmath.zeta(-1, 1, 1))*(2*pi)^(1/4)*2^(5/24)

Formula

Equals exp(1/24)*(2*Pi)^(1/4)*2^(5/24)/A^(5/2) where A = A074962.

Equals exp(-1/6+(5/2)*zeta'(-1))*(2*Pi)^(1/4)*2^(5/24).