Fix the positive integer numbers t1,t2,t3,t4,t5. We have the following formula:
S=∑i,j,h,m,k1+k2+k3+k4=i−t1,ℓ1+ℓ2+ℓ3=j−t2+k4,u1+u2=h−t3+k3+ℓ3M1.M2.M3.M4, where
M_1 = \binom{t_5-k_1}{k_1}\binom{t_4-k_2}{k_2}\binom{t_3-k_3}{k_3}\binom{t_2-k_4}{k_4} M_2 = \binom{t_5-k_1-\ell_1}{\ell_1}\binom{t_4-k_2-\ell_2}{\ell_2}\binom{t_3-k_3-\ell_3}{\ell_3} M_3 = \binom{t_5-k_1-\ell_1-u_1}{u_1}\binom{t_4-k_2-\ell_2-u_2}{u_2}; M_4=\binom{t_1+t_2+t_3+t_4 +t_5-i - j-h-m}{m - t_4 + k_2+ \ell_2 + u_2}.\lambda_i\lambda_j\lambda_h\lambda_m\lambda_{t_1+t_2 + t_3+t_4+t_5 - i - j-h-m} Here, the binomial factors \binom{n}{k} mod 2 and the value of S mod 2. By convention, \binom{n}{k} \equiv 0 (mod 2) if either k < 0 or n < 0 or k > n.
I don't how to construct this formula in SAGE. Can someone show me how to compute it using SAGE?
