Triangle Oak

Studies of Boolean functions
sequences related to seals

Triangles Oak_(Drop)

🌊 triangle Oak (A022166) row sums Daisy (A006116)
k n	0	1	2	3	4	5	6	7	sums
0	1								1
1	1	1							2
2	1	3	1						5
3	1	7	7	1					16
4	1	15	35	15	1				67
5	1	31	155	155	31	1			374
6	1	63	651	1395	651	63	1		2825
7	1	127	2667	11811	11811	2667	127	1	29212

💧 triangle OakDrop (~A289537) row sums Dahlia (~A182176)
k n	0	1	2	3	4	5	6	7	sums
0	1								1
1	0	1							1
2	0	2	1						3
3	0	4	6	1					11
4	0	8	28	14	1				51
5	0	16	120	140	30	1			307
6	0	32	496	1240	620	62	1		2451
7	0	64	2016	10416	11160	2604	126	1	26387

(a, d) ↦ seals

Oak(a, d)
OakDrop(a, d)

is the number of seals with

arity
adicity

a and depth d.

This triangle is the basis of many others. See Triangle Oak with multiplied columns.

The matrix inverse of Oak is SignedTurnedRhododendron.

formulas

$\mathrm {Oak} (n,k)~~=~~\mathrm {Oak} (n-1,k-1)~+~2^{k}\cdot \mathrm {Oak} (n-1,k)$
${\begin{matrix}\mathrm {Oak} (n,k)&~~=~~&\mathrm {Oak} (n-1,k-1)&~+~&2^{k}&\cdot &\mathrm {Oak} (n-1,k)\\\\\mathrm {Oak} (5,3)&~~=~~&\mathrm {Oak} (4,2)&~+~&2^{3}&\cdot &\mathrm {Oak} (4,3)\\155&~~=~~&35&~+~&8&\cdot &15\end{matrix}}$

relationship to Birch

\mathrm {Oak} (n,k)~~=~~\sum _{i=0}^{k}\mathrm {Birch} (n-i,n-k)\cdot {\binom {n}{i}}

${\begin{matrix}\mathrm {Oak} (n,k)&~~=~~&\sum _{i=0}^{k}\mathrm {Birch} (n-i,n-k)\cdot {\binom {n}{i}}\\\\\mathrm {Oak} (5,3)&~~=~~&\sum _{i=0}^{3}\mathrm {Birch} (5-i,5-3)\cdot {\binom {5}{i}}\\155&~~=~~&\mathrm {Birch} (5,2)\cdot {\binom {5}{0}}~+~\mathrm {Birch} (4,2)\cdot {\binom {5}{1}}~+~\mathrm {Birch} (3,2)\cdot {\binom {5}{2}}~+~\mathrm {Birch} (2,2)\cdot {\binom {5}{3}}\\155&~~=~~&40\cdot 1~+~13\cdot 5~+~4\cdot 10~+~1\cdot 10\\\end{matrix}}$

Triangles ExOak_(Drop)

Oak with columns multiplied by powers of two

🌊 triangle ExOak (~A289537) row sums ShortDahlia (A182176)
k n	0	1	2	3	4	5	6	7	sums
0	1 · 1 1								1
1	1 · 1 1	1 · 2 2							3
2	1 · 1 1	3 · 2 6	1 · 4 4						11
3	1 · 1 1	7 · 2 14	7 · 4 28	1 · 8 8					51
4	1 · 1 1	15 · 2 30	35 · 4 140	15 · 8 120	1 · 16 16				307
5	1 · 1 1	31 · 2 62	155 · 4 620	155 · 8 1240	31 · 16 496	1 · 32 32			2451
6	1 · 1 1	63 · 2 126	651 · 4 2604	1395 · 8 11160	651 · 16 10416	63 · 32 2016	1 · 64 64		26387
7	1 · 1 1	127 · 2 254	2667 · 4 10668	11811 · 8 94488	11811 · 16 188976	2667 · 32 85344	127 · 64 8128	1 · 128 128	387987

OakDrop with columns multiplied by powers of two

💧 triangle ExOakDrop row sums ShortDahliaDrop
k n	0	1	2	3	4	5	6	7	sums
0	1 · 1 1								1
1	0	1 · 2 2							2
2	0	2 · 2 4	1 · 4 4						8
3	0	4 · 2 8	6 · 4 24	1 · 8 8					40
4	0	8 · 2 16	28 · 4 112	14 · 8 112	1 · 16 16				256
5	0	16 · 2 32	120 · 4 480	140 · 8 1120	30 · 16 480	1 · 32 32			2144
6	0	32 · 2 64	496 · 4 1984	1240 · 8 9920	620 · 16 9920	62 · 32 1984	1 · 64 64		23936
7	0	64 · 2 128	2016 · 4 8064	10416 · 8 83328	11160 · 16 178560	2604 · 32 83328	126 · 64 8064	1 · 128 128	361600

Padded Oak with rows multiplied by powers of two

💧 triangle ExOakDrop row sums ShortDahliaDrop
k n	0	1	2	3	4	5	6	7	sums
0	1 · 1 1								1
1	0	1 · 2 2							2
2	0	1 · 4 4	1 · 4 4						8
3	0	1 · 8 8	3 · 8 24	1 · 8 8					40
4	0	1 · 16 16	7 · 16 112	7 · 16 112	1 · 16 16				256
5	0	1 · 32 32	15 · 32 480	35 · 32 1120	15 · 32 480	1 · 32 32			2144
6	0	1 · 64 64	31 · 64 1984	155 · 64 9920	155 · 64 9920	31 · 64 1984	1 · 64 64		23936
7	0	1 · 128 128	63 · 128 8064	651 · 128 83328	1395 · 128 178560	651 · 128 83328	63 · 128 8064	1 · 128 128	361600

(a, d) ↦ seal blocks

ExOak(a, d)
ExOakDrop(a, d)

is the number of seal blocks with

arity
adicity

a and depth d.

illustrated examples

_OakDrop (seals with adicity 5) ExOak (blocks with arity 4)

OakDrop(5, 0)=0
seals with arity 5 and depth 0 Oak(5, 0) = 1 with adicity 0 OakDrop(5, 0) = 0 with adicity 5

OakDrop(5, 1)=ExOak(4, 4)=1 ⋅ 6=16
seals with arity 5 and depth 1 Oak(5, 1) = 31 with adicities 1...5 OakDrop(5, 1) = 16 with adicity 5	seal blocks with arity 4 and depth 4 ExOak(4, 4) = ExOakDrop(4, 4) = 16 with adicity 4

OakDrop(5, 2)=ExOak(4, 3)=15 ⋅ 8=120
seals with arity 5 and depth 2 Oak(5, 2) = 155 with adicities 2...5 OakDrop(5, 2) = 120 with adicity 5	seal blocks with arity 4 and depth 3 ExOak(4, 3) = 120 with adicities 3...4 ExOakDrop(4, 3) = 112 with adicity 4

OakDrop(5, 3)=ExOak(4, 2)=35 ⋅ 4=140
seals with arity 5 and depth 3 Oak(5, 3) = 155 with adicities 3...5 OakDrop(5, 3) = 140 with adicity 5	seal blocks with arity 4 and depth 2 ExOak(4, 2) = 140 with adicities 2...4 ExOakDrop(4, 2) = 112 with adicity 4

OakDrop(5, 4)=ExOak(4, 1)=15 ⋅ 2=30
seals with arity 5 and depth 4 Oak(5, 4) = 31 with adicities 4...5 OakDrop(5, 4) = 30 with adicity 5	seal blocks with arity 4 and depth 1 ExOak(4, 1) = 30 with adicities 1...4 ExOakDrop(4, 1) = 16 with adicity 4

OakDrop(5, 5)=ExOak(4, 0)=1 ⋅ 1=1
seals with arity 5 and depth 5 Oak(5, 5) = OakDrop(5, 5) = 1 with adicity 5	seal blocks with arity 4 and depth 0 ExOak(4, 0) = 1 with adicity 0 ExOakDrop(4, 0) = 0 with adicity 4

triangles

These are the same triangles as shown in the boxes above. They refer to the seals with arity 5.
The main content of each image is a refinement of the shown slice of pyramid Ivy.
Each column for a valency is refined into columns for each equivalence class.
Shown above them are a slice of pyramid Liana and its refinement. Their entries are the column sums of the triangles below.
Triangle OakDrop is still called Maple in these images.

depth 1

depth 2

depth 3

depth 4

Triangles Oak(Drop)

formulas

relationship to Birch

Triangles ExOak(Drop)

illustrated examples

Triangles Oak_(Drop)

Triangles ExOak_(Drop)