Given a sequence (a_k) = a₀, a₁, a₂,... of real numbers, define a new sequence L(a_k) = (b_k) where b_k = a_k² - a_k-1 a_k+1. So (a_k) is log-concave if and only if (b_k) is a nonnegative sequence. Call (a_k) infinitely log-concave if Lⁱ(a_k) is nonnegative for all i ≥ 1. Boros and Moll conjectured that the rows of Pascal's triangle are infinitely log-concave. Using a computer and a stronger version of log-concavity, we prove their conjecture for the nth row for all n ≤ 1450. We also use our methods to give a simple proof of a recent result of Uminsky and Yeats about regions of infinite log-concavity. We investigate related questions about the columns of Pascal's triangle, q-analogues, symmetric functions, real-rooted polynomials, and Toeplitz matrices. In addition, we offer several conjectures.

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