What are the coefficients of the polynomial expansion of binomial coefficients?

Question

Considering the binomial coefficient: $$ {{x}\choose{k} } = \frac{x(x-1)...(x-k+1)}{k!} $$ It is clear that it could be written as a polynomial in $x$ but there are explicit computations of ${{x}\choose{k} }$ as a polynomial $$ \sum_{i=0}^k a_{i,k}x^i $$ i.e. it is possible to compute in general $a_{i,k}$? Thanks for the suggestions!

Answer 1

The Stirling number of the first kind $s(k, p) $ is defined as the coefficient of $x^p$ in the numerator of the polynomial you are asking about. Thus the full expansion is $$\sum_p\frac{s(k, p)} {k!} x^p$$