# What is the growth rate of the products of binomial coefficients?

Question 1: Are the following empirically observed relationships true

$${n \choose 1^a}{n \choose 2^a}{n \choose 3^a}\cdots {n \choose m^a} \sim \exp\bigg(\frac{2n^{1 + \frac{1}{a}}}{a+3}\bigg)$$ where $$a$$ is a fixed positive integer and $$m = \lfloor n^{1/a}\rfloor$$.

$${n \choose b}{n \choose 2b}{n \choose 3b}\cdots {n \choose mb} \sim \exp\bigg(\frac{n^{2}}{2b}\bigg)$$ where $$b$$ is a fixed positive integer and $$m = \lfloor n/b\rfloor$$.

Question 2: What is the growth rate of

$${n \choose 1^ab}{n \choose 2^ab}{n \choose 3^ab}\cdots {n \choose m^ab}$$

For $$a = 3$$, the $$\%$$ error between the asymptotic and the actual product is shown below. We observe that the error is small and is decreasing with $$n$$?

Note: Posted in MO since it was unanswered in MSE. • what does Stirling's approximation say here? (I understand there are too many factorials) – Venkataramana Aug 12 '19 at 3:25
• in the second one you probably have $k$ and $m$ confused – Brendan McKay Aug 12 '19 at 7:20
• @BrendanMcKay corrected – Nilotpal Kanti Sinha Aug 12 '19 at 7:53

## 1 Answer

Partial answer (but the general case should be similar): using Stirling+Euler-MacLaurin+Glaisher's constant we have for $$a=1$$: $$\prod_{1\le j\le n}\binom{n}{j}\sim n^{-(n/2+1/3)}e^{n^2/2+n(1-\log(2\pi)/2)+K}$$ with $$K=1/12-2\zeta'(-1)-\log(2\pi)/2\;.$$ Thus I presume that $$A\sim B$$ in the OP's question means that $$\log(A)/\log(B)$$ tends to 1.