1985 Putnam 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.

通过 $f_0(x)=1$, $f_n(0)=0$ 定义 $n \geq 0$ 的多项式 $f_n(x)$

对于 $n \geq 1$，并且

$$

\frac{d}{dx} f_{n+1}(x) = (n+1)f_n(x+1)

$$

对于 $n \geq 0$。有证据地找出以下的显式因式分解

$f_{100}(1)$ 转换为不同素数的幂。