Putnam 1980 B6

Problem B-6 An infinite array of rational numbers $G(d, n)$ is defined for integers $d$ and $n$ with $1 \leqslant d \leqslant n$ as follows: $$G(1, n)=\frac{1}{n}, G(d, n)=\frac{d}{n} \sum_{i=d}^{n} G(d-1, i-1) \text { for } d>1 .$$

For $1<d \leqslant p$ and $p$ prime, prove that $G(d, p)$ is expressible as a quotient $s / t$ of integers $s$ and $t$ with $t$ not an integral multiple of $p$. (For example, $G(3,5)=7 / 4$ with the denominator 4 not a multiple of 5 .)