The log-convexity of a class of linear recurrence sequences

In this paper, we consider the sequences \( \{F(n, k)\}_{n \geq k} \) (\(k \geq 1\)) defined by\( F(n, k) = (n – 2)F(n – 1, k) + F(n – 1, k – 1), \quad F(n, 1) = \frac{n!}{2}, \quad F(n, n) = 1. \) We mainly study the log-convexity of \( \{F(n, k)\}{n \geq k} \) (\(k \geq 1\)) when \( k \) is fixed. We prove that \( \{F(n, 3)\}{n \geq 3}, \{F(n, 4)\}{n \geq 5}, \) and \( \{F(n, 5)\}{n \geq 6} \) are log-convex. In addition, we discuss the log-behavior of some sequences related to \( F(n, k) \).
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