a(n) := f(n, n), where f(0,0) = 1/3, f(0,k) = 0 and f(k,0) = 3^(k-1) if k > 0, and f(n, m) = f(n, m-1) + f(n-1, m) + 3*f(n-1, m-1) otherwise.

A369580

a(n) := f(n, n), where f(0,0) = 1/3, f(0,k) = 0 and f(k,0) = 3^(k-1) if k > 0, and f(n, m) = f(n, m-1) + f(n-1, m) + 3*f(n-1, m-1) otherwise.