Averaged Multinomial Coefficient

Question

Following on from the asymptotic value of the central binomial coefficient, namely:

$$\dbinom{2n}{n}\sim\dfrac{4^n}{\sqrt{\pi n}}$$

we have the multinomial coefficient:

$$\dbinom{n}{k_1 k_2\dots k_m}=\dfrac{n!}{k_1!k_2!\dots k_m!}$$

Is there an asymptotic value for the 'averaged multinomial coefficient', which I call:

$$\dbinom{nk}{n_1 \dots n_k}=\dfrac{(nk)!}{n!^k}$$

I tried using Stirling's approximation:

$$n!\sim\sqrt{2\pi n}\big(\dfrac{n}{e}\big)^n$$

but it's not very pretty.

Answer 1

Yes, it follows from Stirling. For fixed positive integer $k$, as $n \to \infty$,

$$ \dfrac{(nk)!}{n!^k} \sim (2 \pi n)^{(1-k)/2} k^{nk + 1/2} $$