Triangle Rhododendron

Studies of Boolean functions
sequences related to Boolean functions
Triangle Oak with multiplied columns

Triangles Rhododendron(Drop)

Oak with columns multiplied by Chrysanthemum

🌊    triangle Rhododendron    row sums Ranunculus (A028361)
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 3 · 1 3 1 · 2 2 6
3 1 · 1 1 7 · 1 7 7 · 2 14 1 · 8 8 30
4 1 · 1 1 15 · 1 15 35 · 2 70 15 · 8 120 1 · 64 64 270
5 1 · 1 1 31 · 1 31 155 · 2 310 155 · 8 1240 31 · 64 1984 1 · 1024 1024 4590
6 1 · 1 1 63 · 1 63 651 · 2 1302 1395 · 8 11160 651 · 64 41664 63 · 1024 64512 1 · 32768 32768 151470
7 1 · 1 1 127 · 1 127 2667 · 2 5334 11811 · 8 94488 11811 · 64 755904 2667 · 1024 2731008 127 · 32768 4161536 1 · 2097152 2097152 9845550

OakDrop with columns multiplied by Chrysanthemum.

💧    triangle RhododendronDrop    row sums RanunculusDrop
k
n
 0 1 2 3 4 5 6 7 sums
0 1 * 1 1 1
1   0 1 * 1 1 1
2   0 2 * 1 2 1 * 2 2 4
3   0 4 * 1 4 6 * 2 12 1 * 8 8 24
4   0 8 * 1 8 28 * 2 56 14 * 8 112 1 * 64 64 240
5   0 16 * 1 16 120 * 2 240 140 * 8 1120 30 * 64 1920 1 * 1024 1024 4320
6   0 32 * 1 32 496 * 2 992 1240 * 8 9920 620 * 64 39680 62 * 1024 63488 1 * 32768 32768 146880
7   0 64 * 1 64 2016 * 2 4032 10416 * 8 83328 11160 * 64 714240 2604 * 1024 2666496 126 * 32768 4128768 1 * 2097152 2097152 9694080

The main part of RhododendronDrop is Rhododendron with rows multiplied by powers of two.

Triangles TurnedRhododendron(Drop)

signed triangles
base
triangle 		families BF
diagonal triangle total diagonal triangle total
strong Turned
Rhodo. 		Thyme SignedBerberis Aloe Grass SignedBramble ExAloe
balanced & strong Saffron SignedArdisia Agave Sesame SignedAronia ExAgave
dense Pascal Thyme SignedCypress Lily Grass SignedCedar Lotus
balanced & dense Saffron SignedOlearia Chamomile Sesame SignedOleander Calendula

TurnedRhododendron is the base triangle of some signed triangles, making it part of the definition of some sequences, e.g. Aloe.

🌊    triangle TurnedRhododendron    row sums Ranunculus (A028361)
k
n
 0 1 2 3 4 5 6 7 sums
0 1 1
1 1 1 2
2 2 3 1 6
3 8 14 7 1 30
4 64 120 70 15 1 270
5 1024 1984 1240 310 31 1 4590
6 32768 64512 41664 11160 1302 63 1 151470
7 2097152 4161536 2731008 755904 94488 5334 127 1 9845550
💧    triangle TurnedRhododendronDrop    row sums RanunculusDrop
k
n
 0 1 2 3 4 5 6 7 sums
0 1 1
1 0 1 1
2 1 2 1 4
3 6 11 6 1 24
4 56 106 63 14 1 240
5 960 1864 1170 295 30 1 4320
6 31744 62528 40424 10850 1271 62 1 146880
7 2064384 4097024 2689344 744744 93186 5271 126 1 9694080

Triangle SignedRhododendron

The row sums of positive entries are TornRanunculus.

triangle SignedRhododendron (A158474)
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 3 · 1 −3 1 · 2 2 0
3 1 · 1 1 7 · 1 −7 7 · 2 14 1 · 8 −8 0
4 1 · 1 1 15 · 1 −15 35 · 2 70 15 · 8 −120 1 · 64 64 0
5 1 · 1 1 31 · 1 −31 155 · 2 310 155 · 8 −1240 31 · 64 1984 1 · 1024 −1024 0
6 1 · 1 1 63 · 1 −63 651 · 2 1302 1395 · 8 −11160 651 · 64 41664 63 · 1024 −64512 1 · 32768 32768 0
7 1 · 1 1 127 · 1 −127 2667 · 2 5334 11811 · 8 −94488 11811 · 64 755904 2667 · 1024 −2731008 127 · 32768 4161536 1 · 2097152 −2097152 0

SignedTurnedRhododendron is the matrix inverse of Oak.

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