Concavity Properties of Numbers Satisfying the Binomial Recurrence

We consider square arrays of numbers \(\{a(n, k)\}\), generalizing the binomial coefficients:
\(a(n, 0) = c_n\), where the \(c_n\) are non-negative real numbers; \(a(0, k) = c_0\), and if \(n, k > 0\), then \(a(n, k) = a(n, k – 1) + a(n – 1, k)\).
We give generating functions and arithmetical relations for these numbers. We show that every row of such an array is eventually log concave, and give a few sufficient conditions for columns to be eventually log concave. We also give a necessary condition for a column to be eventually log concave, and provide examples to show that there exist such arrays in which no column is eventually log concave.