Factorizations of an n-cycle into two n-cycles

We prove by elementary combinatorial methods that the number of factorizations of an n-cycle (with n odd) into the product of two n-cycles is 2(n−1)!/(n+1). Moreover, we generalize our method to the factorization of an even permutation in Sn into the product of two n-cycles, and we present an algorithm giving all the factorizations of any odd permutation in Sn+1 into the product of an (n+1)-cycle and an n-cycle, where the fixed element of the second permutation is given.