<s>[INST] Coefficient Problem (polynomial expansion) 
Let $C$ be the coefficient of $x^2$ in the expansion of the product $(1 - x)(1 + 2x)(1 - 3x)\cdots(1 + 14x)(1 - 15x).$ Find $|C|.$

Just to begin,
$(1-x)(1+2x) = -2x^2 + x + 1$
$(1-x)(1+2x)(1-3x) = 6x^3 - 5x^2 - 2x + 1$ 
But expanding on like this take too long.
In the end the terms will be like:
$(1 - 15x)(a_0 + a_1x^1 + a_2x^2 + ....)$
Then just looking at the $x^2$, it will be:
$(a_2x^2 - 15a_1x^2) = x^2(a_2 - 15a_1)$
But it it still a very hard problem, any hints?
 [/INST] Somewhat more generally, let $$\prod_{j=1}^n (1 + (-1)^j j x) = 1 + c(n) x + d(n) x^2 + \ldots$$
so you want $d(15)$.
We have $c(0) = d(0) = 0$ with
$$ (1 + c(n-1) x + d(n-1) x^2)(1 + (-1)^n n x) = 1 + c(n) x + d(n) x^2 + \ldots $$
so that 
$$ \eqalign{c(n) &= c(n-1) + (-1)^n n \cr
            d(n) &= d(n-1) + (-1)^n n c(n-1) \cr} $$
Can you solve these recursions?
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