Define $$f_n(x_1,x_2,x_3,x_4)=\sum_{s_1=0}^n a_{n,s_1,s_1,s_1,s_1} x_1^{s_1}x_2^{s_1}x_3^{s_1}x_4^{s_1}+\sum_{s_i\neq s_j, \forall i\neq j,s_i\in[0,n]} a_{n,s_1,s_2,s_3,s_4} x_1^{s_1}x_2^{s_2}x_3^{s_3}x_4^{s_4} $$ where $x_i\in (0,1)$ $\forall i$ and $$a_{n,s_1,s_2,s_3,s_4} =(-1)^{s_1+s_2+s_3+s_4}\binom{n}{s_1}\binom{n}{s_2}\binom{n}{s_3}\binom{n}{s_4}\binom{n+s_1}{s_1}\binom{n+s_2}{s_2} $$
I need a closed form for $f_n(x_1,x_2,x_3,x_4)$
We have the first sum as $$\sum_{s_1=0}^n a_{n,s_1,s_1,s_1,s_1} x_1^{s_1}x_2^{s_1}x_3^{s_1}x_4^{s_1}=\sum_{s_1=0}^n \binom{n}{s_1}^4\binom{n+s_1}{s_1}^2 (x_1x_2x_3x_4)^{s_1}$$ So we get using Wolfram alpha
$$\sum_{s_1=0}^n a_{n,s_1,s_1,s_1,s_1} x_1^{s_1}x_2^{s_1}x_3^{s_1}x_4^{s_1}= \,_6F_5(-n,-n,-n,-n,n+1,n+1;1,1,1,1,1;x_1x_2x_3x_4)$$ Any help would be highly appreciated.