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Thank you for answer! If we take $s_{n,\ell,m}(x)=t_{n,\ell,m}(x)+s_{n,\ell-1,m}(m-\ell+1)-t_{n,\ell,m}(m-\ell+1), t_{n,\ell,m}(x)=\int (n-\ell)^2 s_{n-1,\ell,m}(x)\,dx, s_{n,0,m}(x)=n!x^n$, then $s_{n,n,n}(0)$ is A204264. Is it possible to get something like $R(n,q)$ here?
@PeterTaylor, thank you for comment! Nice conjecture! Could you share how you came to this conclusion? Are there non-recursive methods to compute $A(x)$?
