Why does the exponential generating function approach fail here?

When you have two exponential generating functions $\displaystyle f(x) = \sum_{n=0}^\infty a_n \frac{x^n}{n!}$ and $\displaystyle g(x) = \sum_{n=0}^\infty b_n\frac{x^n}{n!}$, their product $\displaystyle f(x)g(x) = \sum_{n=0}^\infty c_n\frac{x^n}{n!}$ has $\displaystyle c_n = \sum_{k=0}^n \binom{n}{k}a_kb_{n-k}$.

The problem is that each two-unit block is just that: a two-unit block. If I want an $n$-unit-tall tower containing $k$ two-unit blocks and $n-2k$ one-unit blocks (no colors at the moment), there are only $n-k$ total blocks and thus $\displaystyle \binom{n-k}{k}$ ways to arrange them, not $\displaystyle \binom{n}{k}$ ways as would be computed in the product of generating functions.