It is well known that combinatorial identities of Rogers-Ramanujan type arise naturally from certain specializations of characters of integrable highest weight modules for affine Lie algebras. It is also known that for any positive integer k, the integrable highest weight module L(kΛ0) for an (untwisted) affine Lie algebra ĝ has a natural structure of a vertex operator algebra. In this paper using certain results for vertex operator algebras we obtain certain recurrence relations for the characters of L(kΛ0). In the case when k = 1 and ĝ is of (ADE)-type, we solve these recurrence relations, obtaining the full characters of L(kΛ0). Then taking the principal specialization we obtain new families of multisum identities of Rogers-Ramanujan type.