Split Equally

For any arrangement of signs, let $P$ be the set of numbers given the symbol $+$ , and let $N$ be the set of numbers given the symbol $-$ . Then $P$ and $N$ are disjoint subsets of $\{2,3,...,64\}$ whose union is $\{2,3,...,64\}$ , and $1 + \sum_{n \in P}n + \sum_{n \in N}n \; = \; 1 + 2 + \cdots + 64 = 2080 \hspace{2cm} 1 + \sum_{n \in P}n - \sum_{n \in N}n \; = \; 0$ and hence $1 + \sum_{n \in P}n \; = \; \sum_{n \in N}n \; = \; 1040$ and hence we need to count the number of subsets $P$ of $\{2,3,...,64\}$ such that $\sum_{n \in P}n = 1039$ . Thus we want the coefficient of $x^{1039}$ in the product $\prod_{k=2}^{64}(1+ x^k)$ The Mathematica command

SeriesCoefficient[Product[1 + x^k,{k,2,64}],{x,0,1039}]

gives us the answer $\boxed{24435006625667338}$ .

Here's a C++ code of mine to it. Simply bottom-up dynamic programming.

Click here for code