We give a new proof of formulae for the generating series of (Hodge) genera of symmetric products X (n) with coefficients, which hold for complex quasi-projective varieties X with any kind of singularities and which include many of the classical results in the literature as special cases. Important specializations of our results include generating series for extensions of Hodge numbers and Hirzebruch’s χ y -genus to the singular setting and, in particular, generating series for intersection cohomology Hodge numbers and Goresky-MacPherson intersection cohomology signatures of symmetric products of complex projective varieties. Our proof applies to more general situations and is based on equivariant Kunneth formulae and pre-lambda structures on the coefficient theory of a point, $${\bar{K}_0(A(pt))}$$ , with A(pt) a Karoubian $${\mathbb{Q}}$$ -linear tensor category. Moreover, Atiyah’s approach to power operations in K-theory also applies in this context to $${\bar{K}_0(A(pt))}$$ , giving a nice description of the important related Adams operations. This last approach also allows us to introduce very interesting coefficients on the symmetric products X (n).