Partition function P

Built-in:

say partitions(6666)   # very fast

User-defined:

func partitionsP(n) {
    func (n) is cached {

        n <= 1 && return n

        var a = sum(1..floor((sqrt(24*n + 1) + 1)/6), {|k|
            (-1)**(k-1) * __FUNC__(n - ((k*(3*k - 1)) >> 1))
        })

        var b = sum(1..ceil((sqrt(24*n + 1) - 7)/6), {|k|
            (-1)**(k-1) * __FUNC__(n - ((k*(3*k + 1)) >> 1))
        })

        a + b
    }(n+1)
}

var t = Time.micro

say partitionsP.map(0..25).join(' ')
say partitionsP(6666)

say ("Took %.4f seconds" % Time.micro-t)

Output:

1 1 2 3 5 7 11 15 22 30 42 56 77 101 135 176 231 297 385 490 627 792 1002 1255 1575 1958
193655306161707661080005073394486091998480950338405932486880600467114423441282418165863
Took 24.5225 seconds