View source for Triangle Birch

{{DISPLAYTITLE:Triangle ''Birch''}}
{{Seal sequences header}}

[[File:OEISicon.svg|15px]] ~{{oeis|A139382}}

{{Seal triangle Birch}}

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

{{multiple image
 | align = left  | total_width = 800
 | image1 = Number pyramid Liana; adicity minus valency is 0.png
 | image2 = Number pyramid Ivy; adicity minus valency is 0.png
 | footer = <abbr title="arity or adicity">''a''</abbr> = <abbr title="valency">''v''</abbr> in pyramids [[Template:Seal pyramids Liana and Ivy|''Liana'' and ''Ivy'']]
}}
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* T(n,k) = (2^k-1) * T(n-1,k) + T(n-1,k-1)
* second diagonal: Eulerian numbers {{oeislink|A000295}} &nbsp; a(n) = 2^n - n - 1  &nbsp; <small>(see also {{oeislink|A125128}}, {{oeislink|A130103}})</small>
* column 2: {{oeislink|A003462}} &nbsp; a(n) = (3^n - 1)/2
* column 3: {{oeislink|A016212}} &nbsp; a(n) = (7^(n+2) - 3^(n+3) + 2)/24

==formulas==

{{Seal sequences/formulas/Birch}}

===relationship to [[Triangle Oak|''Oak'']]===

{{Seal sequences/formulas/Birch and Oak}}

==Triangles ''TwistedBirch<sub>(Drop)</sub>''==

{{Triangle pair TwistedBirch}}

TwistedBirch(''n'', ''k'') is the number of seals with '''arity''' ''n'' and depth ''k'', whose gravity is 0.<br>
<span style="opacity: .5;">TwistedBirchDrop(''n'', ''k'') is the number of seals with '''adicity''' ''n'' and depth ''k'', whose gravity is 0.</span>

{{multiple image
 | align = left  | total_width = 800
 | image1 = Number pyramid TwistedLiana; gravity 0.png
 | image2 = Number pyramid Lonicera; gravity 0.png
 | footer = gravity 0 in pyramids [[Template:Seal pyramids TwistedLiana and Lonicera|''TwistedLiana'' and ''Lonicera'']]
}}

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The third diagonal of ''TwistedBirchDrop'' is 0, 3, 12, 39, 120, 363... That seems to be <math>a(n) = \frac{3^n - 3}{2}</math> ({{oeis|A029858}}).<br>
The equivalent in ''TwistedBirch'' is higher by 1: &nbsp; 1, 4, 13, 40, 121, 364... That seems to be <math>a(n) = \frac{3^n - 1}{2}</math> ({{oeis|A003462}}).

The fourth diagonal in ''TwistedBirch'' is 1, 11, 90, 670, 4811... That seems to be {{oeis|A016212}}.

==Triangle ''Betula''==

{{Triangle Betula}}

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

[[Category:Seal triangles/pages]]