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 {(n1, n2); 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 (1/2)n(5/6)n (2)n(5/3)n . 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 lowertriangular 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.