Factorization of alternating sums of Virasoro characters

G. Andrews proved that if n is a prime number then the coefficients ak and ak+n of the product (q,q)∞/(qn,qn)∞=∑kakqk have the same sign, see [G. Andrews, On a conjecture of Peter Borwein, J. Symbolic Comput. 20 (1995) 487–501]. We generalize this result in several directions. Our results are based on the observation that many products can be written as alternating sums of characters of Virasoro modules.