let $b\_{n,k}$ be the numbers defined formally by $$X^n=\sum\_{k=0}^n b\_{n,k}\binom{X}{k}$$ where $\binom{X}{n}=\frac{1}{n!}\prod\_{k=0}^{n-1}(X-k)$.


I am looking for an equivalent of $b\_{n,k}$ when $k$ is fixed and $n\to+\infty$.


Thanks in advance for any answer.



Equivalent of Stirling-like numbers
You are looking for a scaled version of the [Stirling numbers of the second kind](https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind):


Note that $\binom{X}{n} = \frac{(x)\_n}{n!}$ where $(x\_n)$ is the falling factorial (or Pochhammer symbol).


