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

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