Consider a uniform random walk on $\mathbb{R}$, with stepsize chosen uniformly from the interval $(-1,1)$. The random walk start at $x=0$. Denote by $\rho_p dx$ the probability that the random walk returns to the interval $(0,dx)$ after $p$ steps, for infinitesimally small $dx$. For $p=1$ and for $p=2$ one has $\rho_p=1/2$, for $p=3$ one has $\rho_p=3/8$, for large $p$ the diffusion approximation gives $\rho_p=(2\pi p/3)^{-1/2}$.

**Q:** Is $\rho_p$ known exactly for any $p$?

*Motivation:* this would give the small-$a$ limit of the box integral in https://mathoverflow.net/a/331993/11260 , $\lim_{a\rightarrow 0}a^{1-p}f_p(a)=2^p\rho_p$, see also fedja's comment in https://math.stackexchange.com/q/3233160/87355