Pascal's Triangle

Pascal's Triangle is defined recursively as:

if row = 0 or col = 0 or row = col then 1
else P(row, col) = P(row - 1, col - 1) + P(row - 1, col)

The two elements on the row above and columns directly adjacent to the current cell are added giving us a new element.

    0 | 1
    1 | 1 1
    2 | 1 2 1
    3 | 1 3 3 1
    4 | 1 4 6 4 1
       ----------
        0 1 2 3 4

    P (2,1) = P(1,0) + P(1,1) = 1 + 1 = 2

In tacit J:

    pasc=: (-&1 1 +&$: -&1 0)`1:@.(=/ +. 0&=)
    pasc 2 1
2

An alternative way of calculating pascal's triangle is to use the iteration operator:

    pasc2=: 3 : '(0&, + ,&0)^:(i.y)1'
    pasc2 5
1 0 0 0 0
1 1 0 0 0
1 2 1 0 0
1 3 3 1 0
1 4 6 4 1

Using the binomial coefficient is the best way, following directly from the definition of pascal's triangle:

    pasc3=: [: !/~ i.
    pasc3 5
1 1 1 1 1
0 1 2 3 4
0 0 1 3 6
0 0 0 1 4
0 0 0 0 1

An interesting property of pascal's triangle is that the diagonals add up to the fibonacci numbers:

    </. pasc3 5
+-+---+-----+-------+---------+-------+-----+---+-+
|1|1 0|1 1 0|1 2 0 0|1 3 1 0 0|4 3 0 0|6 1 0|4 0|1|
+-+---+-----+-------+---------+-------+-----+---+-+

    ({. [: +//. pasc3) 10
1 1 2 3 5 8 13 21 34 55