Exact Dyadic Periods for a Family of Generalized Central Trinomial Coefficients

Abstract

We study the generalized central trinomial family B(n; u, v) = Σ from j = 0 to n of C(n, j)² uⁿ⁻ʲvʲ for integral parameters u and v. For parameters of opposite parity, for every k ≥ 4 and all n ≥ 0, B(n + 2ᵏ⁻²; u, v) ≡ B(n; u, v) modulo 2ᵏ; the shift 2ᵏ⁻² is a period from n = 0. When the even parameter has 2-adic valuation 1, the period 2ᵏ⁻² is least. For every k ≥ 4, the sequence aₙ = B(n; −1, 2), OEIS A098332, has least period 2ᵏ⁻² modulo 2ᵏ. The same specialization also yields a truncated 3-adic congruence governed by Kummer carries. For all four series of Gonçalves and Zagier and every n ≥ 0, no spectral coefficient equals 1/4.