This notebook gives the derivation of the formula of the coefficient for the simple monomial which appears in both Sections 2 and 4 of the paper .   The first derivation is via the algorithm described in Section 2,  
    and the second via the algorithm described in Section 4.

B[u_, v_]  :=  u! /(v! (u - v)!)

A  = 
      B[4 t,  t + 2] B[4 t,  t - 1] B[3 t  - 2,  t] B[3 t  + 1,  
            t + 1] B[2 t  - 2,  t - 1] B[2 t,  t + 1]  - 
        B[4 t,  t + 2] B[4 t,  t - 1] B[3 t  - 2, t] B[3 t + 1,  
            t + 1] B[2 t  - 2,  t - 1] B[2 t, t]  -  
        B[4 t,  t + 2]  B[4 t,  t - 1] B[3 t  - 2,  t] B[
            3 t  + 1,  t] B[2 t  - 2,  t  - 1] B[2 t + 1,  
            t + 2]  +  
        B[4 t,  t + 2]  B[4 t,  t - 1] B[3 t  - 2,  t] B[
            3 t  + 1,  t] B[2 t  - 2,  t  - 1] B[2 t + 1,  
            t + 1]  +  
        B[4 t,  t + 2]  B[4 t,  t - 1] B[3 t  - 2,  t] B[
            3 t  + 1,  t - 1] B[2 t  - 2,  t  - 1] B[2 t + 2,  
            t + 2]   -  
        B[4 t,  t + 2]  B[4 t,  t - 1] B[3 t  - 2,  t] B[
            3 t  + 1,  t - 1] B[2 t - 2,  t  - 1] B[2 t + 2,  
            t + 1]  -  
        B[4 t,  t + 1] B[4 t,  t] B[3 t - 1,  t + 1] B[3 t, t] B[
            2 t  - 2,  t - 1] B[2 t,  t + 1]  +  
        B[4 t, t + 1] B[4 t, t] B[3 t - 1,  t + 1] B[3 t, t] B[
            2 t  - 2, t - 1] B[2 t, t]  +  
        B[4 t,  t + 1] B[4 t,  t] B[3 t  - 1,  t + 1] B[3 t,  
            t - 1] B[2 t  - 2, t - 1] B[2 t + 1,  t + 2]  -  
        B[4 t,  t + 1] B[4 t,  t] B[3 t  - 1,  t + 1] B[3 t,  
            t - 1] B[2 t  - 2, t - 1] B[2 t + 1,  t + 1 ]  +  
        B[4 t,  t + 1] B[4 t,  t] B[3 t  - 1,  t] B[3 t,  
            t + 1] B[2 t  - 1, t] B[2 t - 1,  t ]   -  
        B[4 t,  t + 1] B[4 t,  t] B[3 t  - 1,  t] B[3 t,  
            t + 1] B[2 t  - 1, t] B[2 t - 1,  t  - 1]  -  
        2 B[4 t,  t + 1] B[4 t,  t] B[3 t  - 1,  t] B[3 t,  
            t] B[2 t  - 1, t] B[2 t,  t ]  +  
        3  B[4 t,  t + 1] B[4 t,  t] B[3 t  - 1,  t] B[3 t,  
            t] B[2 t  - 1, t] B[2 t,  t  - 1]  - 
B[
            4 t,  t + 1] B[4 t,  t] B[3 t  - 1,  t] B[3 t,  
            t - 1] B[2 t  - 1, t] B[2 t + 1,  t  - 1]   -  
        B[4 t,  t + 1] B[4 t,  t - 1] B[3 t - 1, t + 1] B[3 t + 1,  
            t - 1] B[2 t  - 2, t - 1] B[2  t  + 2, t + 2]  +  
        B[4 t,  t + 1] B[4 t,  t - 1] B[3 t - 1, t + 1] B[3 t + 1,  
            t - 1] B[2 t  - 2, t - 1] B[2 t  + 2, t + 1]   -  
        B[4 t,  t + 1] B[4 t,  t - 1] B[3 t - 1, t] B[3 t + 1,  
            t + 1] B[2 t  - 1, t] B[2  t , t + 1]  +  
        3  B[4 t,  t + 1] B[4 t,  t - 1] B[3 t - 1, t] B[
            3 t + 1,  t + 1] B[2 t  - 1, t] B[2 t , t]  -  
        B[4 t,  t + 1] B[4 t,  t - 1] B[3 t - 1, t] B[3 t + 1,  
            t + 1] B[2 t  - 1, t] B[2 t , t - 1]  -  
        2 B[4 t,  t + 1] B[4 t,  t - 1] B[3 t - 1, t] B[3 t + 1,  
            t] B[2 t  - 1, t] B[2 t  + 1 , t]  +  
        B[4 t,  t + 1] B[4 t,  t - 1] B[3 t - 1, t] B[3 t + 1,  
            t] B[2 t  - 1, t] B[2 t  + 1 , t - 1]  +  
        B[4 t, t] B[4 t,  t + 1] B[3 t, t] B[3 t  - 1,  t + 1] B[
            2 t,  t] B[2 t  - 2,  t - 1]  -  
        2 B[4 t, t] B[4 t,  t + 1] B[3 t, t] B[3 t  - 1,  t] B[
            2 t,  t] B[2 t  - 1,  
            t]  +  
B[4 t, t]^2 B[3 t, t]^2 B[
              2 t, t]^2

Y  =  Simplify[A]

Y /. {t  ->  2}

Z  =  
      B[4 t,  2 t]^2  B[2  t, t]^2 ( 
              B[2 t,  t - 1]  -  B[ 2 t, t])^2  -  
        2 B[4 t,  2 t] B[4 t,  2 t  - 1] B[2 t  + 1,  t] B[
            2 t  - 1, 
            t] ( B[2 t,  t - 1]  -  B[ 2 t, t])^2  +  
        B[4 t,  2 t] B[4 t,  2 t  - 2] B[2 t  - 2,  t - 1] ( 
            B[2 t,  t - 1]  -  
              B[ 2 t, 
                t]) (B[2 t,  t + 1] B[2 t  + 2,  t + 2]  -  
              B[2 t,  t] B[2 t  + 2,  t + 1])  +  
        B[4 t,  2 t  + 1] B[4 t,  2 t  - 2] B[2 t  - 1,  
            t  - 1] B[2 t  - 2,  
            t - 1] (B[ 2 t  + 2,  t + 1]  -  
              B[2 t  + 2,  t + 2]) ( 
            B[2 t + 1,  t - 1]  -  B[ 2 t + 1, t])  +  
        B[4 t,  2 t  - 2]^2  B[2 t  - 2,  t - 1]^2 ( 
              B[2 t  + 2,  t]  -  
                B[2 t + 2,  t + 1] )^2

Simplify[Z]