**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.