Triangle Ash

Studies of Boolean functions
sequences related to seals

Pascal's triangle with multiplied columns

Compare triangle Sloe.

Triangles Ash and Aspen

Both diagonals are Aster.

Pascal with columns multiplied by Aster

🌊 triangle Ash row sums Daisy (A006116)
k n	0	1	2	3	4	5	6	7	sums
0	1 · 1 1								1
1	1 · 1 1	1 · 1 1							2
2	1 · 1 1	2 · 1 2	1 · 2 2						5
3	1 · 1 1	3 · 1 3	3 · 2 6	1 · 6 6					16
4	1 · 1 1	4 · 1 4	6 · 2 12	4 · 6 24	1 · 26 26				67
5	1 · 1 1	5 · 1 5	10 · 2 20	10 · 6 60	5 · 26 130	1 · 158 158			374
6	1 · 1 1	6 · 1 6	15 · 2 30	20 · 6 120	15 · 26 390	6 · 158 948	1 · 1330 1330		2825
7	1 · 1 1	7 · 1 7	21 · 2 42	35 · 6 210	35 · 26 910	21 · 158 3318	7 · 1330 9310	1 · 15414 15414	29212

PascalDrop with columns multiplied by Aster

💧 triangle Aspen row sums Dahlia (~A182176)
k n	0	1	2	3	4	5	6	7	sums
0	1 · 1 1								1
1	0	1 · 1 1							1
2	0	1 · 1 1	1 · 2 2						3
3	0	1 · 1 1	2 · 2 4	1 · 6 6					11
4	0	1 · 1 1	3 · 2 6	3 · 6 18	1 · 26 26				51
5	0	1 · 1 1	4 · 2 8	6 · 6 36	4 · 26 104	1 · 158 158			307
6	0	1 · 1 1	5 · 2 10	10 · 6 60	10 · 26 260	5 · 158 790	1 · 1330 1330		2451
7	0	1 · 1 1	6 · 2 12	15 · 6 90	20 · 26 520	15 · 158 2370	6 · 1330 7980	1 · 15414 15414	26387

Ash(n, k) is the number of seals with arity n and valency k.
Aspen(n, k) is the number of seals with adicity n and valency k.

These triangles are sums of the pyramids Liana and Ivy. (Those also have the axis depth from back to front.)
In the images the leftmost column is not shown.

Triangles TwistedAsh and Alder

🌊 triangle TwistedAsh row sums Daisy (A006116)
k n	0	1	2	3	4	5	6	7	sums
0	1								1
1	1	1							2
2	2	2	1						5
3	6	6	3	1					16
4	26	24	12	4	1				67
5	158	130	60	20	5	1			374
6	1330	948	390	120	30	6	1		2825
7	15414	9310	3318	910	210	42	7	1	29212

💧 triangle Alder row sums Dahlia (~A182176)
k n	0	1	2	3	4	5	6	7	sums
0	1								1
1	0	1							1
2	1	1	1						3
3	4	4	2	1					11
4	20	18	9	3	1				51
5	132	106	48	16	4	1			307
6	1172	818	330	100	25	5	1		2451
7	14084	8362	2928	790	180	36	6	1	26387

TwistedAsh(n, k) is the number of seals with arity n and gravity k.
Alder(n, k) is the number of seals with adicity n and gravity k.

The left columns are Aster and AsterDrop.

The first diagonals of Alder are ones, natural numbers, square numbers, cubic minus square numbers (A045991). (Same as in TurnedSloeDrop.)

These triangles are sums of the pyramids TwistedLiana and Lonicera. (Those also have the axis depth from back to front.)

Triangle SignedAsh

triangle SignedAsh row sums Geranium (A323842)
k n	0	1	2	3	4	5	6	7	sums
0	1 · 1 1								1
1	1 · 1 −1	1 · 1 1							0
2	1 · 1 1	2 · 1 −2	1 · 2 2						1
3	1 · 1 −1	3 · 1 3	3 · 2 −6	1 · 6 6					2
4	1 · 1 1	4 · 1 −4	6 · 2 12	4 · 6 −24	1 · 26 26				11
5	1 · 1 −1	5 · 1 5	10 · 2 −20	10 · 6 60	5 · 26 −130	1 · 158 158			72
6	1 · 1 1	6 · 1 −6	15 · 2 30	20 · 6 −120	15 · 26 390	6 · 158 −948	1 · 1330 1330		677
7	1 · 1 −1	7 · 1 7	21 · 2 −42	35 · 6 210	35 · 26 −910	21 · 158 3318	7 · 1330 −9310	1 · 15414 15414	8686