Let $E(n,k)$ denote how many set of $k$ distinct non-negative integer are there such that their sum is an even number $\leq n$.
Let $O(n,k)$ denote how many set of $k$ distinct non-negative integer are there such that their sum is an odd number $\leq n$.
Find all values of $n,k$ such that $E(n,k)=O(n,k)$.
My attempts so far: for $k=1$ only when $n$ is even. For $k=2$ it never works.
Conjecture: $k=1,n$ even are the only possibility.
Thank you for your help.