OFFSET
1,3
COMMENTS
Just as Stirling's formula for the asymptotic expansion of n! involves the constant sqrt{2 Pi}, the asymptotic expansion of the product of all binomial coefficients in a row of Pascal's triangle involves a constant, the reciprocal of the constant C defined and evaluated here.
From Bernd C. Kellner, Oct 13 2024: (Start)
It turns out that 1/C is not the complete asymptotic constant for the product of the binomial coefficients in a row of Pascal's triangle. A constant factor of (2*Pi)^(-1/4) was overlooked in the asymptotic expansion of that product given by Hirschhorn in 2013. The correct asymptotic constant is A377023.
However, the constant C equals the constant F(1) as introduced before in Kellner 2009. The constants F(1), F(2), ... occur in the same 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(1)) = 1/12 + log(2*Pi)/4 - 2*log(A) = gamma/12 - t*zeta(3)/360 with some t in (0,1), and so F(1) lies in the interval (1.0457...,1.0492...) (see Kellner 2009 and 2024). (End)
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..10000
Michael D. Hirschhorn, On the asymptotic behavior of Product_{k=0..n} C(n,k), Fib. Q., 51 (2013), 163-173.
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)^(1/12)*(2*Pi)^(1/4))/A^2 where A denotes the Glaisher-Kinkelin constant.
Equals exp(2*zeta'(-1)-1/12)*(2*Pi)^(1/4).
A closely related constant is K = Product_{n>=1} (n!*(e/n)^(n+1/2))/ ((1+1/(n+1/2))^(1/12)*sqrt(2*Pi*e)) = (2^(1/6)*(3*e)^(1/12)*Pi^(1/4))/A^2 = exp(2*zeta'(-1)-1/12)*2^(1/6)*3^(1/12)*Pi^(1/4) = 1.082293504658977773529439... - Peter Luschny, Jun 22 2012
The sqrt of the constant equals Limit_{n>=1} (Product_{k=1..n-1} k!) / f(n) where f(n) = (2*Pi)^(n/2-1/8)*exp(1/24-3/4*n^2)*n^(1/2*n^2-1/12). - Peter Luschny, Jun 23 2012
EXAMPLE
1.04633506677050318098095065697776037101974218113264442441587534042035751563744...
MAPLE
exp(2*Zeta(1, -1)-1/12)*(2*Pi)^(1/4); evalf(%, 100); # Peter Luschny, Jun 22 2012
MATHEMATICA
RealDigits[(Exp[1]^(1/12) (2 Pi)^(1/4))/Glaisher^2, 10, 100][[1]] (*Peter Luschny, Jun 22 2012 *)
PROG
(Sage)
import mpmath
mpmath.mp.pretty=True; mpmath.mp.dps = 200 #precision
mpmath.exp(2*mpmath.zeta(-1, 1, 1)-1/12)*(2*pi)^(1/4) # Peter Luschny, Jun 22 2012
(PARI) exp(2*zeta'(-1)-1/12)*(2*Pi)^(1/4) \\ Charles R Greathouse IV, Dec 12 2013
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Michael David Hirschhorn, Jun 04 2012
STATUS
approved