import Data.Ratio

pascal :: [(Integer, Rational, Integer)] -> [(Integer, Rational, Integer)]
pascal [] = []
pascal [x] = [x]
pascal ((p1,b1,k1) : (p2,b2,k2) : ps) = (p1 + p2,b1,k1) : pascal ((p2,b2,k2) : ps)

bernoulli' :: [(Integer, Rational, Integer)] -> Integer -> Integer -> Rational
bernoulli' pascalBernK x n
  | x == n = let ((_,result,_):_) = pascalBernK in result
  | otherwise = bernoulli' (pascal tmp) (x+1) n
    where tmp = (1, sum [(p%1) * b / ((k-x-2)%1) | (p, b, k) <- pascalBernK], x+1) : pascalBernK

bernoulli :: Integer -> Rational
bernoulli = bernoulli' [(1, 1%1, 0)] 0