Sequence Aster

Studies of Boolean functions
sequences related to seals

	(a)	(a, d)	(a, v, d, g)	(v, d, g)	(v, d)	(v, g)	(v, g=0)	(v)
smallest house size	Valerian	Juniper	Syringa	Peach	Hawthorn	TurnedFir	Gardenia	Ambrosia
GCD of house sizes	Nepeta	Thuja	Buddleja	Apricot	Buckthorn	TurnedSpruce	Gladiolus	Akanthus
Buddleja · Cydonia = Jacaranda	Salvia	Buxus	Cydonia	Quince	Holly	SlopedGinger	Ginger	Aconite
number of seals	Daisy	Oak	Jacaranda	Cherry	Birch	TurnedPine	Geranium	Aster
number of rooms	Calluna	Yew	Petrea	SlopedMagnolia	Lime	SlopedLavender	Lavender	Heather
number of houses	Clover	Elm	Wisteria	SlopedMagnolia	Lime	SlopedLavender	Lavender	Heather

These sequences appear in triangle Ash and related triangles.
Triangles ^Ash/_Aspen are Pascal's triangle with columns multiplied by Aster (so Aster is their diagonal).
Triangles ^TwistedAsh/_Alder have Aster_(Drop) as their left columns.

Compare sequence Ambrosia.

Sequences Aster_(Drop)

	0	1	2	3	4	5	6	7
🌊 Aster (A135922)	1	1	2	6	26	158	1330	15414
💧 AsterDrop	1	0	1	4	20	132	1172	14084

The row sums of Birch and TwistedBirch are Aster.
The row sums of TwistedBirchDrop are AsterDrop.

Aster is the inverse binomial transform of Daisy.

triangle Birch

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

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

triangles Pine_(Drop)

Pascal with columns multiplied by Geranium

🌊 triangle Pine row sums Aster
k n	0	1	2	3	4	5	6	7	8	9	sums
0	1 · 1 1										1
1	1 · 1 1	1 · 0 0									1
2	1 · 1 1	2 · 0 0	1 · 1 1								2
3	1 · 1 1	3 · 0 0	3 · 1 3	1 · 2 2							6
4	1 · 1 1	4 · 0 0	6 · 1 6	4 · 2 8	1 · 11 11						26
5	1 · 1 1	5 · 0 0	10 · 1 10	10 · 2 20	5 · 11 55	1 · 72 72					158
6	1 · 1 1	6 · 0 0	15 · 1 15	20 · 2 40	15 · 11 165	6 · 72 432	1 · 677 677				1330
7	1 · 1 1	7 · 0 0	21 · 1 21	35 · 2 70	35 · 11 385	21 · 72 1512	7 · 677 4739	1 · 8686 8686			15414
8	1 · 1 1	8 · 0 0	28 · 1 28	56 · 2 112	70 · 11 770	56 · 72 4032	28 · 677 18956	8 · 8686 69488	1 · 152191 152191		245578
9	1 · 1 1	9 · 0 0	36 · 1 36	84 · 2 168	126 · 11 1386	126 · 72 9072	84 · 677 56868	36 · 8686 312696	9 · 152191 1369719	1 · 3632916 3632916	5382862

PascalDrop with columns multiplied by Geranium

💧 triangle PineDrop row sums AsterDrop
k n	0	1	2	3	4	5	6	7	8	9	sums
0	1 · 1 1										1
1	0	1 · 0 0									0
2	0	1 · 0 0	1 · 1 1								1
3	0	1 · 0 0	2 · 1 2	1 · 2 2							4
4	0	1 · 0 0	3 · 1 3	3 · 2 6	1 · 11 11						20
5	0	1 · 0 0	4 · 1 4	6 · 2 12	4 · 11 44	1 · 72 72					132
6	0	1 · 0 0	5 · 1 5	10 · 2 20	10 · 11 110	5 · 72 360	1 · 677 677				1172
7	0	1 · 0 0	6 · 1 6	15 · 2 30	20 · 11 220	15 · 72 1080	6 · 677 4062	1 · 8686 8686			14084
8	0	1 · 0 0	7 · 1 7	21 · 2 42	35 · 11 385	35 · 72 2520	21 · 677 14217	7 · 8686 60802	1 · 152191 152191		230164
9	0	1 · 0 0	8 · 1 8	28 · 2 56	56 · 11 616	70 · 72 5040	56 · 677 37912	28 · 8686 243208	8 · 152191 1217528	1 · 3632916 3632916	5137284

Sequence ShortAster

	0	1	2	3	4	5	6	7
Aster (A135922)	1	1	2	6	26	158	1330	15414
ShortAster	1	2	6	26	158	1330	15414	245578
ShortAster − Aster	0	1	4	20	132	1172	14084	230164

Both sequences appear in triangle Larch: Aster is the left column, and ShortAster are the row sums.

row sums of triangle Larch

triangle Larch row sums [[\|ShortAster]]
k n	0	1	2	3	4	5	6	7	8	9	sums
0	1										1
1	1	1									2
2	2	3	1								6
3	6	12	7	1							26
4	26	64	52	15	1						158
5	158	460	476	204	31	1					1330
6	1330	4464	5596	3180	780	63	1				15414
7	15414	58604	86308	61196	20940	2988	127	1			245578
8	245578	1043968	1767972	1505596	666316	141612	11564	255	1		5382862
9	5382862	25327980	48554444	48385900	26414892	7591340	997740	45228	511	1	162700898

relationships

comparison to Ambrosia

🌊
	0	1	2	3	4	5	6	7
Aster	1	1	2	6	26	158	1330	15414
Ambrosia	1	1	2	6	20	78	300	1092
Δ	0	0	0	0	6	80	1030	14322

💧
	0	2	3	4	5	6	7
AsterDrop	1	1	4	20	132	1172	14084
AmbrosiaDrop	1	1	4	14	58	222	792
Δ	0	0	0	6	74	950	13292

Sequences Aster(Drop)

Sequence ShortAster

relationships

Sequences Aster_(Drop)