8. Pascal’s Triangle

P ← ↑ {(0∘,+,∘0)⍣⍵,1}¨∘⍳

Pascal’s triangle [32].

   P 10
1 0  0  0   0   0  0  0 0 0
1 1  0  0   0   0  0  0 0 0
1 2  1  0   0   0  0  0 0 0
1 3  3  1   0   0  0  0 0 0
1 4  6  4   1   0  0  0 0 0
1 5 10 10   5   1  0  0 0 0
1 6 15 20  15   6  1  0 0 0
1 7 21 35  35  21  7  1 0 0
1 8 28 56  70  56 28  8 1 0
1 9 36 84 126 126 84 36 9 1

   10 ↑ (,∘.+⍨⍳10) {+/⍵}⌸ ,P 10
1 1 2 3 5 8 13 21 34 55

The expression (0,x)+(x,0) or its commute, which generates the next set of binomial coefficients, is present in the document that introduced APL\360 in 1967 [33] and the one that introduced J in 1990 [34a]; in Elementary Functions: An Algorithmic Treatment in 1966 [35], in APL\360 User’s Manual in 1968 [36], in Algebra: An Algorithmic Treatment in 1972 [37], in Introducing APL to Teachers in 1972 [38], in An Introduction to APL for Scientists and Engineers in 1973 [39], in Elementary Analysis in 1976 [40], in Programming Style in APL in 1978 [41], in Notation as a Tool of Thought in 1980 [3a], in A Dictionary of APL in 1987 [42a].

∘.!⍨⍳n also computes Pascal’s triangle of order n . It is shorter, but is less interesting algorithmically and historically.