@WillSawin, nice! I would also like to know what is the definition of this combinatorial interpretation in this case? I'm having a little trouble understanding exactly how it works.

@WillSawin, is there a similar combinatorial interpretation for $R(n,0)$ where $R(n,q)=\sum\limits_{j=0}^{f(q)}g(q,j)R(n-1,j)$ with $R(0,q)=h(q)$?

@TomCopeland, thank you for comment! Done.

@PeterTaylor, if you substitute your result into the original formula, is it possible to get a closed form?

@PeterTaylor, thank you again! I'm too lazy to figure it out myself, so I'll try to ask around from familiar experts.

@PeterTaylor, thank you for comment! Nice observation! Could you tell me how to calculate the square root on PARI so that when extracting the coefficient from the function, an integer result is obtained? I tried c^(1/2) but for large $k$ it still turns out to be a decimal fraction.

@PeterTaylor, thank you for comment! Done.