# Are there combinatorial identities for the sum of “equal step sampled binomial coefficients”?

Are there combinatorial identities for the sum of "equal step sampled binomial coefficients" ?

Here "equal step sampled binomial coefficients", I referred to something like:

$${\displaystyle {\binom {6}{0}} + {\binom {6}{2}}+{\binom {6}{4}} +{\binom {6}{6}} }$$

or

$${\displaystyle {\binom {6}{1}} + {\binom {6}{3}}+{\binom {6}{5}} }$$

or in general, something like:

$${\displaystyle {\binom {n}{1}} + {\binom {n}{1 + k}}+ {\binom {n}{1 + 2k}}+...+{\binom {n}{1 + jk}} }$$

for any integer $${j, k}$$ as long as they satisfy: $${1 + jk \leq n}$$,

or another version:

$${\displaystyle {\binom {n}{0}} + {\binom {n}{k}}+ {\binom {n}{2k}}+...+{\binom {n}{jk}} }$$

for any integer $${j, k}$$ as long as they satisfy: $${jk \leq n}$$

My question is, for the above "summation of equally sampled subsets of binomial coefficients", are there known combinatorial identities ?

This isn't necessarily combinatorial, but we can use roots of unity to determine a closed form for $$\sum \limits_{k=0} ^ \infty \binom{n}{kr+a}$$, for fixed $$k, a, r$$, and letting $$\binom{n}{k}=0$$ when $$k>n$$. In general, $$\ \frac{1}{r} \sum \limits _{j=0}^{r-1} \omega ^{-ja} (1+\omega^j)^n=\sum \limits_{k=0} ^{\infty} \binom{n}{a+rk}$$, where $$\omega=e^{2\pi i/r}$$