Let $F_m(x)=\binom{x+m}{m}$. Does anyone know how to prove or disprove that
$$ F_{m-1}(2x)=2^{m-1}F_m(x)+\sum_{d=2}^{\lfloor \frac{m}{2} \rfloor +1} \frac{(-1)^{d-1}}{2} \frac{m2^m}{(d-1)2^{d-1}} \binom{m-d}{d-2}F_{m-d}(x) $$
What I did : I have the checked the identity for $m\leq 10$.
Context : If true, the identity would help me a lot into turning a recent partial answer of mine into a full answer.