The log-balancedness of generalized derangement numbers

Let $$\{d_{n}^{(v)}\}_{n\ge 0}$$ be the sequence of generalized derangement numbers, where v is a nonnegative integer. In this paper, we mainly study the log-balancedness of $$\{d_{n}^{(v)}\}_{n\ge 0}$$ , where $$v\ge 1$$ . We prove that $$\{d_{n}^{(1)}\}_{n\ge 1}$$ and $$\{d_{n}^{(v)}\}_{n\ge 0}$$ ( $$v\ge 2$$ ) are log-balanced. In addition, we discuss the log-balancedness of some sequences involving $$d_{n}^{(v)}$$ . For example, we show that $$\{d_{n+1}^{(1)}-d_{n}^{(1)}\}_{n\ge 1}$$ is log-balanced.