The system has the form

$$ (n-2)f_n^{(1)}=n(f_{n-1}^{(1)}+1), $$ $$ (n-2\cdot 2) f_n^{(2)}=n(f_{n-1}^{(2)}+f_{n-1}^{(1)}), $$ $$ \ldots $$ $$ (n-2k)f_n^{(k)}=n(f_{n-1}^{(k)}+f_{n-1}^{(k-1)}), $$ for the unknown sequences $f_n^{(1)},f_n^{(2)},\ldots,f_n^{(k)}$ with the initial conditions $f^{(i)}_k=0,$ for all $k=0,1,\ldots,2i.$

By direct calculation I have got

$$ f_n^{(1)}=n(n-2), $$ $$ f_n^{(2)}=\frac{1}{2!} (n-4) { n \choose 2} (3n-7), $$ $$ f^{(3)}_n=\frac{1}{3!} (n-6) { n \choose 3} \left( 19{n}^{2}-141n+254 \right), $$ $$ f^{(4)}_n=\frac{1}{4!} (n-8) { n \choose 4} \left( 211{n}^{3}-3258{n}^{2}+16481n-27306 \right), $$ $$ f^{(5)}_n=\frac{1}{5!} (n-10) { n \choose 5} \left( 3651{n}^{4}-96550{n}^{3}+946185{n}^{2}-4071950n+ 6492024 \right) $$

**Question.** What is a general expression for $f_n^{(i)}$?

The ordinary generating function for the above sequences has the form $$ G(f_n^{(1)},z)={\frac {{z}^{3} \left( 3-z \right) }{ \left( 1-z \right) ^{3}}}=3{z}^{3}+8{z}^{4}+15{z}^{5}+\cdots $$ $$ G(f_n^{(2)},z)={\frac {z^5(-3{z}^{3}+16{z}^{2}-35z+40)}{ \left( 1-z \right) ^{5}}}, $$ $$ G(f_n^{(3)},z)={\frac {{z}^{7} \left( -40{z}^{5}+288{z}^{4}-897{z}^{3}+1575{z} ^{2}-1701z+1155 \right) }{ \left( 1-z \right) ^{7}}} $$