Can we consider the formal power series $P(t)=\sum_{m=0}^{\infty} \binom{2m}{m} \left[m^l(4t-1)^l+F_l(t) \right]t^m$ as a formal group law in 1 variable ? Where $l \geq 0$ are integers and $F_l(t)$ are some polynomials in $t$. The first three are $F_1(t)=2t$, $F_2(t)=2t+12t^2$, $F_3(t)=8t^3+20t^2+2t$. It is given that the power series vanishes identically i.e., $P(t)=0 $ for all $t$.
For formal group law $P(0)=0$ must be zero, which obviously satisfies. I can not verify it.