On a conjecture of Ira Gessel

Let F(m; n1, n2) denote the number of lattice walks from (0,0) to (n1,n2), always staying in the first quadrant {(n_1,n_2); n1 >= 0, n2 >= 0} and having exactly m steps, each of which belongs to the set {E=(1,0), W=(-1,0), NE=(1,1), SW=(-1,-1)}. Ira Gessel conjectured that F(2n; 0, 0) = 16^n (1/2)_n (5/6)_n / ((2)_n (5/3)_n) where (a)_n is the Pochhammer symbol. We pose similar conjectures for some other values of (n1,n2), and give closed-form formulas for F(n1; n1, n2) when n1 >= n2 as well as for F(2n2 - n1; n1, n2) when n1 <= n2. In the main part of the paper, we derive a functional equation satisfied by the generating function of F(m; n1, n2), use the kernel method to turn it into an infinite lower-triangular system of linear equations satisfied by the values of F(m; n1, 0) and F(m; 0, n2) + F(m; 0, n2 - 1), and express these values explicitly as determinants of lower-Hessenberg matrices with unit superdiagonals whose non-zero entries are products of two binomial coefficients.