Sequence Thyme

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

${\text{Thyme}}(n)={\frac {2^{2^{n}}+(2^{2^{n-1}})(2^{n}-1)}{2^{n}}}$

Sequences Thyme_(Drop)


	0	1	2	3	4	5	6	7
🌊 Thyme (A000231)	2	3	7	46	4336	134281216	288230380379570176	2658455991569831764110243006194384896
💧 ThymeDrop	2	1	4	39	4290	134276880	288230380245288960	2658455991569831763822012625814814720

a ↦ families

Thyme(a)
ThymeDrop(a)

is the number of families with

a.

row sums of Olive_(Drop)

Oak with columns multiplied by Aloe

🌊 triangle Olive (A227725) row sums Thyme (A000231)
k n	0	1	2	3	4	5	sums
0	1 · 2 2						2
1	1 · 2 2	1 · 1 1					3
2	1 · 2 2	3 · 1 3	1 · 2 2				7
3	1 · 2 2	7 · 1 7	7 · 2 14	1 · 23 23			46
4	1 · 2 2	15 · 1 15	35 · 2 70	15 · 23 345	1 · 3904 3904		4336
5	1 · 2 2	31 · 1 31	155 · 2 310	155 · 23 3565	31 · 3904 121024	1 · 134156284 134156284	134281216

OakDrop with columns multiplied by Aloe

💧 triangle OliveDrop row sums ThymeDrop
k n	0	1	2	3	4	5	sums
0	1 · 2 2						2
1	0	1 · 1 1					1
2	0	2 · 1 2	1 · 2 2				4
3	0	4 · 1 4	6 · 2 12	1 · 23 23			39
4	0	8 · 1 8	28 · 2 56	14 · 23 322	1 · 3904 3904		4290
5	0	16 · 1 16	120 · 2 240	140 · 23 3220	30 · 3904 117120	1 · 134156284 134156284	134276880