Sequence Lotus

Studies of Boolean functions
sequences related to Boolean functions

triangles by arity and strength
	base triangle	diagonal	families		BF
	base triangle	diagonal	triangle	total	triangle	total
all	Oak	(Ex)Aloe	Olive	Thyme	ExOlive	Grass
balanced	Oak	(Ex)Agave	Laurel	Saffron	ExLaurel	Sesame
dense	Birch	(Ex)Aloe	Lemon	Lily	ExLemon	Lotus
b & d	Birch	(Ex)Agave	Orange	Chamomile	ExOrange	Calendula

Lotus are the row sums of the weight triangle A163353. See here.

Sequences Lotus_(Drop)


	0	1	2	3	4	5	6	7
🌊 Lotus (A000371)	2	2	10	218	64594	4294642034	18446744047940725978	340282366920938463334247399005993378250
💧 LotusDrop	2	0	8	208	64376	4294577440	18446744043646083944	340282366920938463315800654958052652272

a ↦ dense

Lotus(a) is the number of dense BF with arity a.

a ↦ gravity 0

Lotus(a)
LotusDrop(a)

is the number of BF with

half


	0	1	2	3	4	5	6	7
🌊 HalfLotus (A003465)	1	1	5	109	32297	2147321017	9223372023970362989	170141183460469231667123699502996689125
💧 HalfLotusDrop	1	0	4	104	32188	2147288720	9223372021823041972	170141183460469231657900327479026326136

row sums of ExLemon

Birch with columns multiplied by ExAloe

triangle ExLemon row sums Lotus (A000371)
k n	0	1	2	3	4	5	sums
0	1 · 2 2						2
1	0 · 2 0	1 · 2 2					2
2	0 · 2 0	1 · 2 2	1 · 8 8				10
3	0 · 2 0	1 · 2 2	4 · 8 32	1 · 184 184			218
4	0 · 2 0	1 · 2 2	13 · 8 104	11 · 184 2024	1 · 62464 62464		64594
5	0 · 2 0	1 · 2 2	40 · 8 320	90 · 184 16560	26 · 62464 1624064	1 · 4293001088 4293001088	4294642034

row sums of SignedCedar

The absolute values are Pascal with columns multiplied by Grass.

triangle SignedCedar row sums Lotus (A000371)
k n	0	1	2	3	4	5	sums
0	1 · 2 2						2
1	1 · 2 −2	1 · 4 4					2
2	1 · 2 2	2 · 4 −8	1 · 16 16				10
3	1 · 2 −2	3 · 4 12	3 · 16 −48	1 · 256 256			218
4	1 · 2 2	4 · 4 −16	6 · 16 96	4 · 256 −1024	1 · 65536 65536		64594
5	1 · 2 −2	5 · 4 20	10 · 16 −160	10 · 256 2560	5 · 65536 −327680	1 · 4294967296 4294967296	4294642034

SmallLotus differs from Lotus only in the first entry.
SmallLotus_(Drop) are the left columns of SmallTurnedWillow_(Drop).


	0	1	2	3	4
🌊 SmallLotus	1	2	10	218	64594
💧 SmallLotusDrop	1	1	8	208	64376

a ↦ gravity 0

SmallLotus(a)
SmallLotusDrop(a)

is the number of BF with

a and gravity 0.