Template:Gender of Boolean functions/triangles/male

♂

This triangle shows the numbers of male Boolean functions with arity $n$ and valency $k$ .
The right diagonal is ${\boldsymbol {q}}(n)=2^{(2^{n}-1)}$ . (That is half the number of all $n$ -ary Boolean functions.) (like A058891, but with offset 0)
This is Pascal's triangle with column $k$ multiplied by ${\boldsymbol {q}}(k)$ .

The row sums are A246418.

k n	0	1	2	3	4	5	sums
0	1 · 1 1						1
1	1 · 1 1	1 · 2 2					3
2	1 · 1 1	2 · 2 4	1 · 8 8				13
3	1 · 1 1	3 · 2 6	3 · 8 24	1 · 128 128			159
4	1 · 1 1	4 · 2 8	6 · 8 48	4 · 128 512	1 · 32768 32768		33337
5	1 · 1 1	5 · 2 10	10 · 8 80	10 · 128 1280	5 · 32768 163840	1 · 2147483648 2147483648	2147648859

from math import factorial


def binom(n, k):
    return factorial(n) // (factorial(k) * factorial(n - k))


for n in range(6):
    row = []
    for k in range(n+1):
        row.append(
            binom(n, k) * 2 ** (2 ** k - 1)
        )
    print(n, '-->', row, '-->', sum(row))

illustrated triangles

These images show the male functions also seen in the two 8×256 matrices. (Only a few of the 128 on the right side of the matrix are shown.)
They are ordered by adicity (vertical) and valency (horizontal).
The numbers sum up to those in the triangle above. (E.g. 2 + 2 + 2 = 6 and 8 + 16 = 24.)

Template:Gender of Boolean functions/triangles/male

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