Partial Binomial Sums

Question

Is there a closed form for the solutions of partial binomial sums of the form

$\displaystyle \sum_{n=0}^{\lfloor\dfrac{j-p}{k}\rfloor} \dbinom{j}{nk+p}$

where $\{j,k,p\}\in \mathbb{R}$?

If not, how about the cases where $k|j$ and $p=0$ or $k|j-p$?

Answer 1

Assuming $0\leq p<n$, then the formula is $$a_j=\frac{\sum_{k=0}^{n-1} \zeta_n^{-pk}(1+\zeta_n^k)^j}{n}$$

where $\zeta_n$ is a primitive $n$th root of unity.

This formula will be dominated by $\frac{2^j}{n}$ - that is $\frac{a_j}{2^j/n}\to 1$.