Series multisection is a folklore formula (Riordan called it an "ancient vintage" in his 1968 book "Combinatorial identities"), which from a given analytical generating function for some numerical sequence allows one to obtain a generating function for a subsequence with indicies forming an arithmetic progression. In particular, it leads to a closed-form expression for sums of binomial coefficients taken with a certain step $c$:
$${q\choose d} + {q\choose d+c} + {q\choose d+2c} + \cdots = \frac{1}{c}\cdot \sum_{k=0}^{c-1} \left(2 \cos\frac{\pi k}{c}\right )^q\cdot \cos \frac{\pi(q-2d)k}{c}.$$