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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)

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).