Exact Ratio and Root-Ratio Monotonicity for Two Binomial-Sum Sequences

For the binomial-sum sequence aₙ = Σₖ₌₀ⁿ binom(n,k)² binom(2k,k)/(2k−1) and its normalized partial sums bₙ = n⁻² Σₖ₌₀ⁿ⁻¹ aₖ, we give exact-index proofs of strict increase of the successive-term ratios to 9 and strict decrease of the successive root ratios to 1. We also prove sharp eventual arithmetic concavity of both successive-term ratio sequences and establish the precise first-order rates at which the ratios approach their common limit: 18/n and 36/n, respectively. The proofs start from a telescoping certificate for the original binomial sum, establish a compatible recurrence for the normalized partial sums, and use rational upper barriers together with explicit polynomial-positivity identities. An elementary log-difference lemma treats the two root-ratio conclusions uniformly. Arithmetic divisibility results previously proved by Guo and Liu are acknowledged separately and are not presented as results of this manuscript. Licensing note: The manuscript is licensed under CC BY 4.0. The OEIS b-file snapshots included in the supplementary archive retain their original CC BY-SA 4.0 license and attribution. The independently authored verification code is made available under CC BY 4.0.