Triangle Birch

Studies of Boolean functions
sequences related to seals

triangle Birch (~A139382) row sums Aster (A135922)
k n	0	1	2	3	4	5	6	7	sums
0	1								1
1	0	1							1
2	0	1	1						2
3	0	1	4	1					6
4	0	1	13	11	1				26
5	0	1	40	90	26	1			158
6	0	1	121	670	480	57	1		1330
7	0	1	364	4811	7870	2247	120	1	15414

(a, d) ↦ seals

Birch(a, d) is the number of seals with arity = adicity = valency = a and depth = d.

(a, s) ↦ BF

Birch(a, d) is the number of BF with arity = valency = a and strength s.

Birch is the basis of triangles (Ex)Lemon and (Ex)Orange. This makes it part of the definition of Lotus and Calendula, which count dense BF.

The matrix inverse of Birch is SignedEukalyptus.

Oak = Pascal ∘ Birch equivalently: Birch = Pascal⁻¹ ∘ Oak

a = v in pyramids Liana and Ivy

T(n,k) = (2^k-1) * T(n-1,k) + T(n-1,k-1)
second diagonal: Eulerian numbers A000295 a(n) = 2^n - n - 1 (see also A125128, A130103)
column 2: A003462 a(n) = (3^n - 1)/2
column 3: A016212 a(n) = (7^(n+2) - 3^(n+3) + 2)/24

formulas

$\mathrm {Birch} (n,k)~~=~~\mathrm {Birch} (n-1,k-1)~+~(2^{k}-1)\cdot \mathrm {Birch} (n-1,k)$
${\begin{matrix}\mathrm {Birch} (n,k)&~~=~~&\mathrm {Birch} (n-1,k-1)&~+~&(2^{k}-1)&\cdot &\mathrm {Birch} (n-1,k)\\\\\mathrm {Birch} (5,3)&~~=~~&\mathrm {Birch} (4,2)&~+~&(2^{3}-1)&\cdot &\mathrm {Birch} (4,3)\\90&~~=~~&13&~+~&7&\cdot &11\end{matrix}}$

relationship to Oak

\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 TwistedBirch_(Drop)

🌊 triangle TwistedBirch (~A139382) row sums Aster (A135922)
k n	0	1	2	3	4	5	6	7	sums
0	1								1
1	1	0							1
2	1	1	0						2
3	1	4	1	0					6
4	1	11	13	1	0				26
5	1	26	90	40	1	0			158
6	1	57	480	670	121	1	0		1330
7	1	120	2247	7870	4811	364	1	0	15414

💧 triangle TwistedBirchDrop row sums AsterDrop
k n	0	1	2	3	4	5	6	7	sums
0	1								1
1	0	0							0
2	0	1	0						1
3	0	3	1	0					4
4	0	7	12	1	0				20
5	0	15	77	39	1	0			132
6	0	31	390	630	120	1	0		1172
7	0	63	1767	7200	4690	363	1	0	14084

(a, d) ↦ seals

TwistedBirch(a, d)
TwistedBirchDrop(a, d)

is the number of seals with

arity
adicity

a and depth d.

gravity 0 in pyramids TwistedLiana and Lonicera

The third diagonal of TwistedBirchDrop is 0, 3, 12, 39, 120, 363... That seems to be $a(n)={\frac {3^{n}-3}{2}}$ (A029858).
The equivalent in TwistedBirch is higher by 1: 1, 4, 13, 40, 121, 364... That seems to be $a(n)={\frac {3^{n}-1}{2}}$ (A003462).

The fourth diagonal in TwistedBirch is 1, 11, 90, 670, 4811... That seems to be A016212.

formulas

relationship to Oak

Triangles TwistedBirch(Drop)

Triangles TwistedBirch_(Drop)