Putnam 1985 B2

Define polynomials $f_n(x)$ for $n \geq 0$ by $f_0(x)=1$, $f_n(0)=0$ for $n \geq 1$, and $$\frac{d}{dx} f_{n+1}(x) = (n+1)f_n(x+1)$$ for $n \geq 0$. Find, with proof, the explicit factorization of $f_{100}(1)$ into powers of distinct primes.