Compositions with position-dependent part restrictions: a proof of a conjecture of Bala

We prove a conjecture of Peter Bala recorded in 2022 in the OEIS entry A348474. Let a(n) denote the number of weak compositions of n into exactly 2n parts, where every positive part in an odd position must be odd, while parts in even positions are unrestricted. Bala conjectured that a(n) = [xⁿ] Φ(x)ⁿ, where Φ(x) = (1 + x − x²) / ((1 + x)(1 − x)²). We establish this formula through a direct generating-function argument based on independent part-slots, and extend the method to arbitrary position-dependent restrictions that are periodic in the index. We also prove the consequences recorded by Bala: two double binomial expressions for a(n), an exponential relation with the lattice-path sequence A133656, and the Gauss congruences a(np^k) ≡ a(np^(k−1)) (mod p^k). In addition, we prove the functional equation for the generating function of A133656, which is recorded in the OEIS without proof or reference.