We consider a balanced urn with $a_{0} = 1, b_{0} = 0, a = d = 0$ and $b = c = 1$. In other words, the urn initially contains one white ball and, at each draw, we add a ball of the opposite color to the one that was drawn. For all real $u$ and $v$, we set $P_{0}(u,v) = u^{a_{0}} v^{b_{0}}$ and $P_{n}(u,v) = \sum_{\omega \in \Omega_{n}} u^{b(\omega)} v^{n(\omega)}$.
Verify that $P_{3}(u,v) = uv^{3} + 4u^{2}v^{2} + u^{3}v$.