Is there a formula for determining the coefficients of $\prod\limits_{i=0}^n(a+i)$

Question

The binomial theorem and Pascal's triangle make it very straight forward to figure out the coefficients for $(a+b)^n$

Is there a similar formula for determining the coefficients for:

$$(a)(a+1)(a+2)*\dots*(a+n)$$

where $a,n>0$ are integers.

Any pointers or hints (if this problem is very elementary) are greatly appreciated.

Answer 1

The (unsigned) Stirling number of the first kind $c(n,k)$ counts the number of permutations of $\{1,2,\dots,n\}$ having $k$ cycles in their disjoint cycle decomposition.

Using the recurrence relation $$ c(n+1,k)=c(n,k-1)+nc(n,k)$$ together with $c(0,0)=1$, it's not hard to show by induction that $$ x(x+1)\cdot (x+n-1)=\sum_{k=0}^nc(n,k)x^k $$

So the coefficients you are after are $c(n+1,k)$.