AUSTRIAN-POLISH 1983 Problem 8

(a) Prove that $(2^{n+1}-1)!$ is divisible by $\prod_{i=0}^n (2^{n+1-i}-1)^{2^i }$ , for every natural number n (b) Define the sequence ( $c_n$ ) by $c_1=1$ and $c_{n}=\frac{4n-6}{n}c_{n-1}$ for $n\ge 2$ . Show that each $c_n$ is an integer.