What is the expected distance of the electron from the nucleus in the hydrogen atom?

$$\langle r\rangle_{n,l,m}=\frac{a_0n^2}{Z}\left[1+\frac{1}{2}\left(1-\frac{l(l+1)}{n^2}\right)\right].$$

Source: McQuarrie, Quantum Chemistry.

There exists something called Kramers's recursion rule and I think it is what are you looking for.

$$\frac{k+1}{n^2} \left\langle r^k \right\rangle - \frac{a_0}{Z} \left(2k+1\right)\left\langle r^{k-1} \right\rangle + \frac{k a_0^2}{4Z^2} \left( \left(2l+1\right)^2 - k^2 \right) \left\langle r^{k-2} \right\rangle,$$

where $k$ is integer and $a_0$ Bohr radius. For deriving $\left\langle r \right\rangle$ you have to calculate $\left\langle r^{-1} \right\rangle$ at first by setting $k=0$ and then you can set $k=1$ and calculate $\left\langle r \right\rangle$. And of course you know $\left\langle r^{0} \right\rangle = 1$.

The result is

$$\left\langle r \right\rangle = \frac{a_0}{2Z}\left(3n^2-l\left(l+1\right)\right).$$