A377023 Decimal expansion of the asymptotic constant of the product of binomial coefficients in a row of Pascal's triangle.

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

Offset: 0

Author

Bernd C. Kellner, Oct 13 2024

Keywords

• nonn
• cons

Comments

The asymptotic product of binomial coefficients in the n-th row of Pascal's triangle as n goes to infinity provides an asymptotic constant C. This constant must lie in the interval [0.590727...,0.631618...), where the interval is derived from asymptotic products of binomial coefficients over the rows. Indeed, the constant C can also be derived from a limiting case of the latter products (see Kellner 2024).

The constant C is involved with a certain constant F(1) = A213080. The constants F(1), 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. By a divergent series expansion, it follows that F(1) lies in the interval (1.0457...,1.0492...) (see Kellner 2009 and 2024).

Examples

			0.60364867603601031967070211804205268306704463040701700740585803621917783756033...
		

Links

• Bernd C. Kellner, Table of n, a(n) for n = 0..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, A377024.

Programs

Maple

exp(1/12-2*Zeta(1, -1))/(2*Pi)^(1/2); evalf(%, 100);

Mathematica

RealDigits[Glaisher^2/(Exp[1/12] (2 Pi)^(1/2)), 10, 100][[1]]

PARI

default(realprecision, 100);
exp(1/12-2*zeta'(-1))/(2*Pi)^(1/2)

Sage

import mpmath
mpmath.mp.pretty = True; mpmath.mp.dps = 100
mpmath.exp(1/12-2*mpmath.zeta(-1, 1, 1))/(2*pi)^(1/2)

Formula

Let A = A074962 denote the Glaisher-Kinkelin constant.

Equals 1/(A213080*(2*Pi)^(1/4)).

Equals A^2/(exp(1/12)*(2*Pi)^(1/2)).

Equals exp(1/12-2*zeta'(-1))/(2*Pi)^(1/2).