Is it possible to simplify this binomial expression?

Question

There is binomial expression(s) written as $$\sum_{n>0}\frac{(-3n+2k-3)n!^2}{2(2n+1)(k-1)!^2(n-k+1)!^2 \binom{2n}{n}}=0\; if\; k=0\; or= -1\; if\; k>0$$ which simplifies to $$\sum_{n>0}\frac{(3n-2k+1)\binom{n}{k}^2}{(2n+1) \binom{2n}{n}}=2\; for\; all\; k>-1$$ Logically this looks like a physical impossibility. Would anyone believe that there is any way this could be true and if so how would one go about proving this paradox or seemingly impossibility.