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A377023
Decimal expansion of the asymptotic constant of the product of binomial coefficients in a row of Pascal's triangle.
3
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
0,1
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, 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
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).
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