Head-Tail Product

Billy flips a fair coin $n$ times. Let $X$ denote the number of heads, and let $Y$ denote the number of tails. $\mathbb{E}[XY]$ can be expressed in the form

$$\begin{aligned} \frac{an^2 + bn + c}{d} \end{aligned}$$

for some integers $a, b, c, d$, where $a, b, c$ are each relatively prime with $d$. Compute $a + b + c + d$.