number of self-conjugate partitions of n (with all parts even)

Question

What is the generating function?

I know the g.f. of number of self-conjugate partitions and the g.f. of the number of partitions with all parts even. But I don't know how to proceed after that.

Any hint is appreciated!

Answer 1

I figure out the solution with the help of Somos.

Just a hint for others: Take use of the Ferrers diagram, and replace each point in the diagram with something else. (e.g. a set of points maybe?)