Take the graded algebra $A=\mathbb C[a_1,a_2,a_3,\dots]$ with grading given by $\deg a_{i}=2i$. Let $a(t)$ be the generating function for generators ($a(t)=a_1t+a_2t^2+\cdots\in A[[t]]$). What is the quotient of $A$ by relations, generating function of which is given by $a(t)^2$ — i.e. is there some nice description of the graded algebra $ \mathbb C[a_1,a_2,a_3,\dots]/(a(t)^2=0):=\mathbb C[a_1,a_2,a_3,\dots]/(a_1^2,2a_1a_2,2a_1a_3+a_2^2,\dots,\sum_{i+j=k}a_ia_j,\dots)? $ What about Hilbert series, at least?
Upd (from off-site communications). Feigin-Stoyanovsky looks relevant: the algebra in question appears in theorem 2.2.1 as some representation of $\widehat{sl}_2$, and formula 2.3.3 tells that Hilbert series is exactly $G(q)$ of Rogers-Ramanujan. Mysterious but interesting.