Triangular Recurrences

Let T be a function of two integers, n and k. T is a triangular recurrence if it satisfies

T(n,k) = f(n,k)T(n+a,k+b) + g(n,k)T(n+c,k+d)

Example 1: Binomial Coefficients
B(n,k) = B(n-1,k-1) + B(n-1,k)

Example 2: Stirling Numbers
S(n,k) = S(n-1,k-1) + kS(n-1,k)

Example 3: Eulerian Numbers
A(n,k) = (n-k)A(n-1,k-1) + kA(n-1,k)