Does anyone know a better expression than the current one for this sum?
$$ \sum_{i=0}^m \binom{N-i}{m-i}, \quad 0 \le m \le N. $$
It would help me compute a lot of things and make equations a lot cleaner in the context where it appears (as some asymptotic expression of the coefficients of a polynomial defined over cyclic graphs). Perhaps the context doesn't help much though.
For instance, if $N = m$, the sum is $N+1$, and for $N = m+1$, this sum is $N + (N-1) + \dots + 1 = N(N+1)/2$. But otherwise I don't know how to compute it.