  For a PI-algebra R over a field of characteristic 0 let T(R) be the T-ideal
of the polynomial identities of R and let c(R,t) be the codimension series of R
(i.e., the generating function of the codimension sequence of R). Let A, B and
R be PI-algebras such that T(R)=T(A)T(B). We show that if c(A,t) and c(B,t) are
rational functions, then c(R,t) is also rational. If c(A,t) is rational and
c(B,t) is algebraic, then c(R,t) is also algebraic. The proof is based on the
fact that the product of two exponential generating functions behaves as the
exponential generating function of the sequence of the degrees of the outer
tensor products of two sequences of representations of symmetric groups.
