On the moments of the Ulam-Kac adder

Let $\{U(n)\}_{n \geq 0}$ be a sequence of independent random variables such that $U(n)$ is distributed uniformly on $\{0, 1, 2 \dots n\}$. The Ulam-Kac adder is the history-dependent random sequence defined by $X_{n + 1} = X_{n} + X_{U(n)}$ with the initial condition $X_0 = 1$. We show that for each $m \geq 1$, it holds that $\log E[X_n^m]/\sqrt{n}$ approaches a constant $c_m$ as $n \to \infty$. Loose bounds are provided for the constants $c_m$.