A challenge

Let $N_k$ be a number of different figures obtained by a putting two Young diagrams of partitions $\la,\mu$, such that $\vert\la\vert+\vert\mu\vert=k$ on top of each other. For example, $N_0=N_1=1$, $N_2=3$, $N_4=5$, $N_5=10$, $N_6=16$.

CHALLENGE. Compute the function $N(t)=\sum_{i=0}^\infty N_it^i$.

At the moment I know the answer but I do not know an elementary proof of it.