Prolog Stirling & Binomial Number Calculator

stirling1(0, 0, 1).
stirling1(_, -1, 0).
stirling1(-1, _, 0).
stirling1(N, K, Res) :- N1 is N - 1, K1 is K - 1, stirling1(N1, K1, Res1), stirling1(N1, K, Res2), Res is Res1 + K * Res2.

stirling2(0, 0, 1).
stirling2(_, -1, 0).
stirling2(-1, _, 0).
stirling2(N, K, Res) :- N1 is N - 1, K1 is K - 1, stirling2(N1, K1, Res1), stirling2(N1, K, Res2), Res is Res1 + N1 * Res2.

binomial(0, 0, 1).
binomial(_, -1, 0).
binomial(-1, _, 0).
binomial(N, K, Res) :- N1 is N - 1, K1 is K - 1, binomial(N1, K1, Res1), binomial(N1, K, Res2), Res is Res1 + Res2.