Moments of Rademacher Sum

Question

Suppose $$S_n = \frac{r_1}{\sqrt n} + \frac{r_2}{\sqrt{n}} + \dots + \frac{r_n}{\sqrt{n}}$$ where $r_i$ are Rademacher (uniform $\pm 1$) variables. Is there a closed form expression for the even moments $$\mathbb{E} S_{n+1}^{m} = \frac{1}{2}\mathbb{E}\biggl(\sqrt{\frac{n}{n+1}}S_n + \tfrac{1}{\sqrt{n+1}}\biggr)^m + \frac{1}{2}\mathbb{E}\biggl(\sqrt{\frac{n}{n+1}}S_n - \tfrac{1}{\sqrt{n+1}}\biggr)^m?$$ In particular, can we show that $$\mathbb{E}S_n^m \leq \mathbb{E} S_{n+1}^m$$ for all $m=1,2,\ldots$ and $n=1,2,\ldots$?