king walk

The Delannoy numbers D(a, b) are the number of lattice paths from (0, 0) to (b, a) in which only east (1, 0), north (0, 1), and northeast (1, 1) steps are allowed (i.e., ->, ↑, and ↗). They are given by the recurrence relation D(a, b) = D(a - 1, b) + D(a, b - 1) + D(a - 1, b - 1), with D(0, 0) = 1. The are also given by the sums D(n, k) | = | sum_(d = 0)^n(k d)(n + k - d k) | = | sum_(d = 0)^n 2^d(k d)(n d) | = | (n + k k)_2 F_1(-n, - k; - (k + n);-1), where _2 F_1(a, b;c;z) is a hypergeometric function.

binomial coefficient | Cantor function | Catalan number | integer sequence primes | Motzkin number | Schmidt's problem | Schröder number