Triangle Willow

Studies of Boolean functions
sequences related to Boolean functions

Pascal's triangle with multiplied columns

Triangles Willow_(Drop)

The diagonal of both triangles is Lotus.

Pascal with columns multiplied by Lotus

🌊 triangle Willow row sums Grass (A001146)
k n	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)
k n	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

(a, v) ↦ all

Willow(a, v)
WillowDrop(a, v)

is the number of BF with

a and valency v.

half

Pascal with columns multiplied by HalfLotus

🌊 triangle HalfWillow row sums HalfGrass (~A058891)
k n	0	1	2	3	4	5	sums
0	1 · 1 1						1
1	1 · 1 1	1 · 1 1					2
2	1 · 1 1	2 · 1 2	1 · 5 5				8
3	1 · 1 1	3 · 1 3	3 · 5 15	1 · 109 109			128
4	1 · 1 1	4 · 1 4	6 · 5 30	4 · 109 436	1 · 32297 32297		32768
5	1 · 1 1	5 · 1 5	10 · 5 50	10 · 109 1090	5 · 32297 161485	1 · 2147321017 2147321017	2147483648

PascalDrop with columns multiplied by HalfLotus

💧 triangle HalfWillowDrop row sums HalfGrassDrop (~A040996)
k n	0	1	2	3	4	5	sums
0	1 · 1 1						1
1	0	1 · 1 1					1
2	0	1 · 1 1	1 · 5 5				6
3	0	1 · 1 1	2 · 5 10	1 · 109 109			120
4	0	1 · 1 1	3 · 5 15	3 · 109 327	1 · 32297 32297		32640
5	0	1 · 1 1	4 · 5 20	6 · 109 654	4 · 32297 129188	1 · 2147321017 2147321017	2147450880

Column 0 is Lotus_(Drop). The row sums without it are NonLotus_(Drop).

🌊 triangle TurnedWillow row sums Grass
k n	0	1	2	3	4	5	sums
0	2						2
1	2	2					4
2	10	4	2				16
3	218	30	6	2			256
4	64594	872	60	8	2		65536
5	4294642034	322970	2180	100	10	2	4294967296

💧 triangle TurnedWillowDrop row sums GrassDrop
k n	0	1	2	3	4	5	sums
0	2						2
1	0	2					2
2	8	2	2				12
3	208	26	4	2			240
4	64376	842	54	6	2		65280
5	4294577440	322098	2120	92	8	2	4294901760

(a, g) ↦ all

TurnedWillow(a, g)
TurnedWillowDrop(a, g)

is the number of BF with

Column 0 is SmallLotus_(Drop). The row sums without it are SmallNonLotus_(Drop).

🌊 triangle SmallTurnedWillow row sums SmallGrass
k n	0	1	2	3	4	5	sums
0	1						1
1	2	1					3
2	10	4	1				15
3	218	30	6	1			255
4	64594	872	60	8	1		65535
5	4294642034	322970	2180	100	10	1	4294967295

💧 triangle SmallTurnedWillowDrop row sums SmallGrassDrop
k n	0	1	2	3	4	5	sums
0	1						1
1	1	1					2
2	8	3	1				12
3	208	26	5	1			240
4	64376	842	54	7	1		65280
5	4294577440	322098	2120	92	9	1	4294901760

(a, g) ↦ all

SmallTurnedWillow(a, g)
SmallTurnedWillowDrop(a, g)

is the number of BF with

a and gravity g. (The contradiction with infinite gravity is not counted.).