OFFSET
0,1
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).
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.
FORMULA
EXAMPLE
0.60364867603601031967070211804205268306704463040701700740585803621917783756033...
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]]
PROG
(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)
(PARI)
default(realprecision, 100);
exp(1/12-2*zeta'(-1))/(2*Pi)^(1/2)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Bernd C. Kellner, Oct 13 2024
STATUS
approved