Triangle Birch

~Template:Oeis

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

• T(n,k) = (2^k-1) * T(n-1,k) + T(n-1,k-1)
• second diagonal: Eulerian numbers Template:Oeislink   a(n) = 2^n - n - 1   (see also Template:Oeislink, Template:Oeislink)
• column 2: Template:Oeislink   a(n) = (3^n - 1)/2
• column 3: Template:Oeislink   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)

TwistedBirch(n, k) is the number of seals with arity n and depth k, whose gravity is 0.
TwistedBirchDrop(n, k) is the number of seals with adicity n and depth k, whose gravity is 0.

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

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

Triangle Betula

Betula is a square array, whose rows are the columns of Birch, but without the empty fields above the diagonal.
The rows are then shortened to get a triangle. (This seems like an arbitrary choice, so the row sums probably do not count anything.)