How to compute the sum $\sum\limits_{k=1}^{\infty} 2k{2k\choose k+1} p^{k-1} (1-p)^{k+1}$

Question

Let $p$ in $(0,1)$. How to compute $$\sum_{k=1}^{\infty} 2k{2k\choose k+1} p^{k-1} (1-p)^{k+1}\ ?$$

Personal tries: I've tried to use generating functions, but I can't deal with this $2 k$ in front of the binomial coefficient ${2k\choose k+1}$.