Template:Gender of Boolean functions/triangles

The number triangles in the following boxes show the numbers of Boolean functions by ^arity/_adicity and valency.
Row indices on the left (🌊) are the arity, on the right (💧) the adicity. Entries on the left are the sums of columns on the right.
These triangles are based on Pascals triangle, with columns multiplied by consecutive entries of a sequence, which becomes the diagonal.

diagonals and row sums

name OEIS 0 1 2 3 4 5 ♀ diag. Violet ~ 2·A007537 1 0 2 90 31826 2147158386 🌊 Primula A246537 1 1 3 97 32199 2147318437 💧 PrimulaDrop 1 0 2 94 32102 2147286238 ♂ diag. HalfGrass ~A058891 1 2 8 128 32768 2147483648 🌊 HalfCrocus A246418 1 3 13 159 33337 2147648859 💧 HalfCrocusDrop 1 2 10 146 33178 2147615522 all diag. Lotus A000371 2 2 10 218 64594 4294642034 🌊 Grass A001146 2 4 16 256 65536 4294967296 💧 GrassDrop ~A111403 2 2 12 240 65280 4294901760 ♂ − ♀ diag. NonLotus A005530 0 2 6 38 942 325262 🌊 Hyacinth 2·A059301 0 2 10 62 1138 330422 💧 HyacinthDrop 0 2 8 52 1076 329284

♀

Pascal with columns multiplied by Violet 🌊    triangle Robinia    row sums Primula (A246537) v a 0 1 2 3 4 5 sums 0 1 · 1 1 1 1 1 · 1 1 1 · 0 0 1 2 1 · 1 1 2 · 0 0 1 · 2 2 3 3 1 · 1 1 3 · 0 0 3 · 2 6 1 · 90 90 97 4 1 · 1 1 4 · 0 0 6 · 2 12 4 · 90 360 1 · 31826 31826 32199 5 1 · 1 1 5 · 0 0 10 · 2 20 10 · 90 900 5 · 31826 159130 1 · 2147158386 2147158386 2147318437 PascalDrop with columns multiplied by Violet 💧    triangle RobiniaDrop    row sums PrimulaDrop v a 0 1 2 3 4 5 sums 0 1 · 1 1 1 1   0 1 · 0 0 0 2   0 1 · 0 0 1 · 2 2 2 3   0 1 · 0 0 2 · 2 4 1 · 90 90 94 4   0 1 · 0 0 3 · 2 6 3 · 90 270 1 · 31826 31826 32102 5   0 1 · 0 0 4 · 2 8 6 · 90 540 4 · 31826 127304 1 · 2147158386 2147158386 2147286238 illustration These images show the pink patterns also seen in the two 8×256 matrices. They are ordered by adicity (vertical) and valency (horizontal). Zhegalkin indices truth tables

♂

Pascal with columns multiplied by HalfGrass 🌊    triangle HalfCedar    row sums HalfCrocus (A246418) v a 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 PascalDrop with columns multiplied by HalfGrass 💧    triangle HalfCedarDrop    row sums HalfCrocusDrop v a 0 1 2 3 4 5 sums 0 1 · 1 1 1 1   0 1 · 2 2 2 2   0 1 · 2 2 1 · 8 8 10 3   0 1 · 2 2 2 · 8 16 1 · 128 128 146 4   0 1 · 2 2 3 · 8 24 3 · 128 384 1 · 32768 32768 33178 5   0 1 · 2 2 4 · 8 32 6 · 128 768 4 · 32768 131072 1 · 2147483648 2147483648 2147615522 illustration These images show the blue patterns 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). Zhegalkin indices truth tables

all

Pascal with columns multiplied by Lotus 🌊    triangle Willow    row sums Grass (A001146) v a 0 1 2 3 4 5 sums 0 1 · 2 2 2 1 1 · 2 2 1 · 2 2 4 2 1 · 2 2 2 · 2 4 1 · 10 10 16 3 1 · 2 2 3 · 2 6 3 · 10 30 1 · 218 218 256 4 1 · 2 2 4 · 2 8 6 · 10 60 4 · 218 872 1 · 64594 64594 65536 5 1 · 2 2 5 · 2 10 10 · 10 100 10 · 218 2180 5 · 64594 322970 1 · 4294642034 4294642034 4294967296 PascalDrop with columns multiplied by Lotus 💧    triangle WillowDrop    row sums GrassDrop (~A111403) v a 0 1 2 3 4 5 sums 0 1 · 2 2 2 1   0 1 · 2 2 2 2   0 1 · 2 2 1 · 10 10 12 3   0 1 · 2 2 2 · 10 20 1 · 218 218 240 4   0 1 · 2 2 3 · 10 30 3 · 218 654 1 · 64594 64594 65280 5   0 1 · 2 2 4 · 10 40 6 · 218 1308 4 · 64594 258376 1 · 4294642034 4294642034 4294901760

♂ − ♀

Pascal with columns multiplied by NonLotus 🌊    triangle Hickory    row sums Hyacinth (2·A059301) v a 0 1 2 3 4 5 sums 0 1 · 0 0 0 1 1 · 0 0 1 · 2 2 2 2 1 · 0 0 2 · 2 4 1 · 6 6 10 3 1 · 0 0 3 · 2 6 3 · 6 18 1 · 38 38 62 4 1 · 0 0 4 · 2 8 6 · 6 36 4 · 38 152 1 · 942 942 1138 5 1 · 0 0 5 · 2 10 10 · 6 60 10 · 38 380 5 · 942 4710 1 · 325262 325262 330422 PascalDrop with columns multiplied by NonLotus 💧    triangle HickoryDrop    row sums HyacinthDrop v a 0 1 2 3 4 5 sums 0 1 · 0 0 0 1   0 1 · 2 2 2 2   0 1 · 2 2 1 · 6 6 8 3   0 1 · 2 2 2 · 6 12 1 · 38 38 52 4   0 1 · 2 2 3 · 6 18 3 · 38 114 1 · 942 942 1076 5   0 1 · 2 2 4 · 6 24 6 · 38 228 4 · 942 3768 1 · 325262 325262 329284

Pascal's triangle
🌊    Pascal k n 0 1 2 3 4 5 sums 0 1 1 1 1 1 2 2 1 2 1 4 3 1 3 3 1 8 4 1 4 6 4 1 16 5 1 5 10 10 5 1 32 💧    PascalDrop k n 0 1 2 3 4 5 sums 0 1 1 1 0 1 1 2 0 1 1 2 3 0 1 2 1 4 4 0 1 3 3 1 8 5 0 1 4 6 4 1 16