Voyage into the golden screen

Question

We start from A004718 named "The Danish composer Per Nørgård's "infinity sequence", invented in an attempt to unify in a perfect way repetition and variation" $$a(2n) = -a(n), a(2n+1) = a(n) + 1, a(0)=0$$ another words $$a_k(n)=(-1)^{n+1}a_k\left(\left\lfloor{n \over k}\right\rfloor\right)+(n \mod k), a_k(0)=0$$ for $k=2$. Next we define $$s_k(n)=\left\lfloor\log_{k}n\right\rfloor, s_k(0)=0$$ also $$p_k(n)=\prod_{i=0}^{s_k(n)} (1+a_k\left(\left\lfloor{n \over k^i}\right\rfloor\right))$$ and finally $$q_k(n) = \sum_{j=0}^{k^n-1}p_k(j)$$ What is nice here, it is the fact, that for even $k$ $$q_k(n)=\binom{k+1}{2}^n$$ How can one prove it? How it can be extended for odd $k$ (I mean simple correction of $a_k(n)$ recurrence relation)?