Digit sums of binomial sums

Let b⩾2 be a fixed positive integer and let S(n) be a certain type of binomial sum. In this paper, we show that for most n the sum of the digits of S(n) in base b is at least , where c0 is some positive constant depending on b and on the sequence of binomial sums. Our results include middle binomial coefficients and Apéry numbers An. The proof uses a result of McIntosh regarding the asymptotic expansions of such binomial sums as well as Bakerʼs theorem on lower bounds for nonzero linear forms in logarithms of algebraic numbers.