Asymptotics of partial sums of central binomial coefficients and Catalan numbers

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We prove exact asymptotic expansions for the partial sums of the sequences of central binomial coefficients and Catalan numbers, ∑ₖ₌₀ⁿ 2kk and ∑ₖ₌₀ⁿ Cₙ. We also obtain closed forms for the polynomials ∑ₖ₌₀q⁻¹2kkxᵏ and ∑ₖ₌₀q⁻¹Cₖxᵏ over the field of p elements, where q is a power of the prime p.