Numerical experiments suggest that
$\binom{2m}{m + k}\cdot\frac{3m - 1 - 2k^2}{2m - 1}$
is integer for all $-m \le k\le +m$. It means that expression evaluation could be implemented very efficiently, only using integer addition and multiplication.

However, I'm failed to derive computationally efficient expression so far.
The two ideas I have are linear combination of binomials, and recursion -- but more insight is needed to go further.


Question: Who are the members of an early design of Luffy's crew?
Answer: I have no good references but:

The mini character was supposed to be the shipwright and was replaced by Franky.

The other one was meant to be a plant obsessed character who was replaced by Robin and largely incorporated into the post time skip Ussop.