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

1,3

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).

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.

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).

Example

1.02393741163711840157795078258621780080376098043644005129469909513476924124007...

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]]

Prog

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

Crossrefs

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

Sequence in context: A021422 A193086 A281289 * A331388 A093319 A160388

Adjacent sequences: A377021 A377022 A377023 * A377025 A377026 A377027

Keyword

nonn,cons

Author

Bernd C. Kellner, Oct 13 2024

Status

approved