For the case in which xn = 0, the inner sum satisfies: min{n1 ,n0 } ) n1 1 − n0 n1 r1 = n1 1 n0 1 r1 − − − r1 =1 r1 n1 r1 1 r1 1 r1 − − 1 r1 =1 min{n1 ,n0 } ) r r1 1 − ,n0 } 1 = n 1 − min{n1 n0 1 n1 − ,n0 } ) − 1 =1 r1 1 n1 − min{n1 = n 1 r1 − − n1 − n0 1 − 1 1 n0 r1 − n1 − 1 ) n0 − 1 r1 =1 ) n0 − = n 1 − 1 n0 1 x − = n 1 − n − n0 1 − 2 n1 n0 1 − n1 − n0 n1 − n1 1 − x min{n1 −1,n0 −1} The left term of the second line follows because it is the total number of sequences that can be formed in the first 1 positions with 1 zeros and = ones. The final line follows n n0 n1 n n0 − − − from an application of Vandermonde’s convolution.44 For the case in which xn = 1, the inner sum can be reduced using similar arguments: convolution is given as Vandermonde’s 44 m + k r + s \ min{r−m,n} = m + n ( \( s \ k=max{−m,n−s} n − ( r k from which one can derive the following identity, which we apply m + k ( min{.£−m,s−n}
For the case in which xn = 0, the inner sum satisfies: min{n1 ,n0 } ) r1 =1 n1 − 1 r1 − 1 n0 n1 − r1 = r1 n1 min{n1 ,n0 } ) r1 =1 n1 − 1 n0 1 − r1 r1 − 1 r1 1 = n − 1 n0 − 1 1 − n1 min{n1 ,n0 } ) r1 =1 min{n1 ,n0 } n1 − 1 r r1 − 1 1 = n − 1 n0 − 1 n0 − n1 ) r1 =1 n1 − 1 r1 − 1 n0 − 1 r1 − 1 = n − 1 n0 − 1 = n − 1 n0 − 1 n0 − n1 n0 − n1 min{n1 −1,n0 −1} ) n − 2 n1 − 1 n1 − 1 x n0 − 1 x The left term of the second line follows because it is the total number of sequences that can be formed in the first n − 1 positions with n0 − 1 zeros and n1 = n − n0 ones. The final line follows from an application of Vandermonde’s convolution.44 For the case in which xn = 1, the inner sum can be reduced using similar arguments: 44 Vandermonde’s convolution is given as min{r−m,n} k=max{−m,n−s} ( r m + k \( s \ n − k ( r + s \ = m + n from which one can derive the following identity, which we apply min{.£−m,s−n} ( m + k