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

$\mathrm{B}\mathrm{i}\mathrm{r}\mathrm{c}\mathrm{h}(n,k)=\mathrm{B}\mathrm{i}\mathrm{r}\mathrm{c}\mathrm{h}(n-1,k-1)+(2^k-1)\cdot \mathrm{B}\mathrm{i}\mathrm{r}\mathrm{c}\mathrm{h}(n-1,k)$

$\begin{array}{ccccccc}\mathrm{B}\mathrm{i}\mathrm{r}\mathrm{c}\mathrm{h}(n,k)& =& \mathrm{B}\mathrm{i}\mathrm{r}\mathrm{c}\mathrm{h}(n-1,k-1)& +& (2^k-1)& \cdot & \mathrm{B}\mathrm{i}\mathrm{r}\mathrm{c}\mathrm{h}(n-1,k)\\ \\ \mathrm{B}\mathrm{i}\mathrm{r}\mathrm{c}\mathrm{h}(5,3)& =& \mathrm{B}\mathrm{i}\mathrm{r}\mathrm{c}\mathrm{h}(4,2)& +& (2^3-1)& \cdot & \mathrm{B}\mathrm{i}\mathrm{r}\mathrm{c}\mathrm{h}(4,3)\\ 90& =& 13& +& 7& \cdot & 11\end{array}$

relationship to Oak

$\mathrm{O}\mathrm{a}\mathrm{k}(n,k)=\sum _{i=0}^k\mathrm{B}\mathrm{i}\mathrm{r}\mathrm{c}\mathrm{h}(n-i,n-k)\cdot (\genfrac{}{}{0ex}{}{n}{i})$

$\begin{array}{ccc}\mathrm{O}\mathrm{a}\mathrm{k}(n,k)& =& \sum _{i=0}^k\mathrm{B}\mathrm{i}\mathrm{r}\mathrm{c}\mathrm{h}(n-i,n-k)\cdot (\genfrac{}{}{0ex}{}{n}{i})\\ \\ \mathrm{O}\mathrm{a}\mathrm{k}(5,3)& =& \sum _{i=0}^3\mathrm{B}\mathrm{i}\mathrm{r}\mathrm{c}\mathrm{h}(5-i,5-3)\cdot (\genfrac{}{}{0ex}{}{5}{i})\\ 155& =& \mathrm{B}\mathrm{i}\mathrm{r}\mathrm{c}\mathrm{h}(5,2)\cdot (\genfrac{}{}{0ex}{}{5}{0})+\mathrm{B}\mathrm{i}\mathrm{r}\mathrm{c}\mathrm{h}(4,2)\cdot (\genfrac{}{}{0ex}{}{5}{1})+\mathrm{B}\mathrm{i}\mathrm{r}\mathrm{c}\mathrm{h}(3,2)\cdot (\genfrac{}{}{0ex}{}{5}{2})+\mathrm{B}\mathrm{i}\mathrm{r}\mathrm{c}\mathrm{h}(2,2)\cdot (\genfrac{}{}{0ex}{}{5}{3})\\ 155& =& 40\cdot 1+13\cdot 5+4\cdot 10+1\cdot 10\end{array}$

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.