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-1a_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 can 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.

This is joint work with Bruce Sagan.

The poster in pdf.