In Pascal's triangle each number is the sum of the two nearest numbers in the line above:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1A three-dimensional analog of Pascal's triangle can be defined as a square pyramid in which each number is the sum of the four nearest numbers in the layer above. Define a function pascalp(n) that returns the nth layer of this pyramid, as follows:
pascalp(1)
1
pascalp(2)
1 1
1 1
pascalp(3)
1 2 1
2 4 2
1 2 1
pascalp(4)
1 3 3 1
3 9 9 3
3 9 9 3
1 3 3 1
pascalp(5)
1 4 6 4 1
4 16 24 16 4
6 24 36 24 6
4 16 24 16 4
1 4 6 4 1Note: Pascal's pyramid can also be defined as a tetrahedron (see http://en.wikipedia.org/wiki/Pascal%27s_pyramid), in which case the layers are triangular rather than square, and the numbers are the trinomial coefficients.
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