Consider a random walk making \(2n\) steps, and let \(T\) be the first
return to its starting point, that is
\[T =\ min\{1 ≤ k ≤ 2n : S_k = 0\},\]
and \(T = 0\) if the walk does not return to zero in the first \(2n steps.

Show that for all \( 1 ≤ k ≤ n\) we have,

\[P(T = 2k) =\frac1{2k − 1} \begin{pmatrix} 2k\\k\end{pmatrix} 2^{-2k}\]

[Meetster Ex 3.3.2]