What is the probability that a random walk starting at 0 will reach +2 in 2 steps, 3 step, 4 steps, etc.?

Question

The random walk I am referring to is a symmetric, unbiased, 1D random walk.

In an answer given in the link below, the probabilities are given for S1, but I am trying to find out what it is for S2, and the more general case (Sn).

Expected number of steps for reaching $K$ in a random walk

Answer 1

This can be done inductively. Suppose that you know $S_n$, with $n>0$. In order to arrive to $n+1$, you first need to arrive to $n$. Due to the loss of memory from the random walk, you can forget everything that happened before arriving to $n$, and start a new random walk. But in this new random walk, you only need one more step to arrive to $n+1$, so the expected number of steps from the first time you arrive at $n$ to arrive at $n+1$ is $S_1$. Since the expected time to arrive to $n$ for the first time is $S_n$, $S_{n+1}=S_n+S_1$.

Moreover, from here we can see that $S_n$ is multiplicative, so $S_n=nS_1$. Using the result from the link you shared, we complete the answer.