OpenLLM-France/wikimedia huggingface dataset:
# Triangle Ash

* Pascal's triangle with multiplied columns
Compare triangle Sloe.

## Triangles Ash and Aspen

Both diagonals are Aster.
🌊    triangle Ash    row sums Daisy (A006116) :
| kn | 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 |
💧    triangle Aspen    row sums Dahlia (~A182176) :
| kn | 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.

## Triangles TwistedAsh and Alder

🌊    triangle TwistedAsh    row sums Daisy (A006116) :
| kn | 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) :
| kn | 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.)
# Economics/History of Economic Thought
## Contents

* Introduction
* Ancient
* Middle Ages
* Pre-classical: Before Adam
* Adam Smith
* Say, Malthus and Ricardo
* Classical Economics
* Socialists Before Marx
* Karl Marx and Marxism; see also: Karl Marx
* The Neo-classical Economics: Marginal Revolution
* Menger and Austrian Economics
* Léon Walras and Vilfredo Pareto: General Equilibrium
* Alfred Marshall
* Historical school and Institutional Economics
* Monetary Theories Before Keynesian Revolution
* John Maynard Keynes and Keynesian Revolution
* Keynesians after Keynes and Neoclassical Synthesis
* Capitalism and Socialism
* Econometrics and Mathematical Economics
* Economists and Policy
* Development of Game Theory
* Milton Friedman and Monetarist Counterrevolution
* Expansion of the Applications of Economic Analysis
* Rational Expectation and After
* Economic Growth Theory and Development Economics
* International Economics
* Postwar Heterodox Economics
* Contemporary Economics
# Triangle Forsythia

* Triangle Oak with multiplied columns

## Triangles Forsythia$_{(Drop)}$

🌊    triangle Forsythia    row sums Dandelion :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | sums |
| 0 | 1 · 1 1 | | | | | | 1 |
| 1 | 1 · 1 1 | 1 · 2 2 | | | | | 3 |
| 2 | 1 · 1 1 | 3 · 2 6 | 1 · 23 23 | | | | 30 |
| 3 | 1 · 1 1 | 7 · 2 14 | 7 · 23 161 | 1 · 3904 3904 | | | 4080 |
| 4 | 1 · 1 1 | 15 · 2 30 | 35 · 23 805 | 15 · 3904 58560 | 1 · 134156284 134156284 | | 134215680 |
| 5 | 1 · 1 1 | 31 · 2 62 | 155 · 23 3565 | 155 · 3904 605120 | 31 · 134156284 4158844804 | 1 · 288230371925149328 288230371925149328 | 288230376084602880 |
💧    triangle ForsythiaDrop    row sums DandelionDrop :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | sums |
| 0 | 1 · 1 1 | | | | | | 1 |
| 1 | 0 | 1 · 2 2 | | | | | 2 |
| 2 | 0 | 2 · 2 4 | 1 · 23 23 | | | | 27 |
| 3 | 0 | 4 · 2 8 | 6 · 23 138 | 1 · 3904 3904 | | | 4050 |
| 4 | 0 | 8 · 2 16 | 28 · 23 644 | 14 · 3904 54656 | 1 · 134156284 134156284 | | 134211600 |
| 5 | 0 | 16 · 2 32 | 120 · 23 2760 | 140 · 3904 546560 | 30 · 134156284 4024688520 | 1 · 288230371925149328 288230371925149328 | 288230375950387200 |
PoplarDrop can be defined with Forsythia and powers of two.
2-padded Forsythia with rows multiplied by powers of two.
💧    triangle PoplarDrop    row sums GrassDrop :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | sums |
| 0 | 2 · 1 2 | | | | | | 2 |
| 1 | 0 | 1 · 2 2 | | | | | 2 |
| 2 | 0 | 1 · 4 4 | 2 · 4 8 | | | | 12 |
| 3 | 0 | 1 · 8 8 | 6 · 8 48 | 23 · 8 184 | | | 240 |
| 4 | 0 | 1 · 16 16 | 14 · 16 224 | 161 · 16 2576 | 3904 · 16 62464 | | 65280 |
| 5 | 0 | 1 · 32 32 | 30 · 32 960 | 805 · 32 25760 | 58560 · 32 1873920 | 134156284 · 32 4293001088 | 4294901760 |
# Sequence Dandelion
## Sequences Dandelion$_{(Drop)}$

${\text{Dandelion}}(n)={\color {Red}2^{2^{n}-n-1}}\cdot ({\color {Green}2^{2^{n}}-1})$  Dandelion = Sage · SmallGrass
🌊 Dandelion :
| 0 | 1 | 1 · 1 |
| 1 | 3 | 1 · 3 |
| 2 | 30 | 2 · 15 |
| 3 | 4080 | 16 · 255 |
| 4 | 134215680 | 2048 · 65535 |
| 5 | 288230376084602880 | 67108864 · 4294967295 |
💧 DandelionDrop :
| 0 | 1 | 1 · 1 |
| 1 | 2 | 2 · 1 |
| 2 | 27 | 9 · 3 |
| 3 | 4050 | 270 · 15 |
| 4 | 134211600 | 526320 · 255 |
| 5 | 288230375950387200 | 4398113617920 · 65535 |
row sums of Forsythia$_{(Drop)}$ :
🌊    triangle Forsythia    row sums Dandelion :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | sums |
| 0 | 1 · 1 1 | | | | | | 1 |
| 1 | 1 · 1 1 | 1 · 2 2 | | | | | 3 |
| 2 | 1 · 1 1 | 3 · 2 6 | 1 · 23 23 | | | | 30 |
| 3 | 1 · 1 1 | 7 · 2 14 | 7 · 23 161 | 1 · 3904 3904 | | | 4080 |
| 4 | 1 · 1 1 | 15 · 2 30 | 35 · 23 805 | 15 · 3904 58560 | 1 · 134156284 134156284 | | 134215680 |
| 5 | 1 · 1 1 | 31 · 2 62 | 155 · 23 3565 | 155 · 3904 605120 | 31 · 134156284 4158844804 | 1 · 288230371925149328 288230371925149328 | 288230376084602880 |
💧    triangle ForsythiaDrop    row sums DandelionDrop :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | sums |
| 0 | 1 · 1 1 | | | | | | 1 |
| 1 | 0 | 1 · 2 2 | | | | | 2 |
| 2 | 0 | 2 · 2 4 | 1 · 23 23 | | | | 27 |
| 3 | 0 | 4 · 2 8 | 6 · 23 138 | 1 · 3904 3904 | | | 4050 |
| 4 | 0 | 8 · 2 16 | 28 · 23 644 | 14 · 3904 54656 | 1 · 134156284 134156284 | | 134211600 |
| 5 | 0 | 16 · 2 32 | 120 · 23 2760 | 140 · 3904 546560 | 30 · 134156284 4024688520 | 1 · 288230371925149328 288230371925149328 | 288230375950387200 |
# Sequence Azalea

| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| padded Daisy | 1 | 1 | 2 | 5 | 16 | 67 | 374 | 2825 |
| powers of two | 1 | 2 | 4 | 8 | 16 | 32 | 64 | 128 |
| Azalea | 1 | 2 | 8 | 40 | 256 | 2144 | 23936 | 361600 |
row sums of Sycamore :
🌊    triangle MapleCore (~A289537)    row sums DahliaCore (A182176) :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | sums |
| 0 | 1 · 1 1 | | | | | | | | 1 |
| 1 | 1 · 1 1 | 1 · 2 2 | | | | | | | 3 |
| 2 | 1 · 1 1 | 3 · 2 6 | 1 · 4 4 | | | | | | 11 |
| 3 | 1 · 1 1 | 7 · 2 14 | 7 · 4 28 | 1 · 8 8 | | | | | 51 |
| 4 | 1 · 1 1 | 15 · 2 30 | 35 · 4 140 | 15 · 8 120 | 1 · 16 16 | | | | 307 |
| 5 | 1 · 1 1 | 31 · 2 62 | 155 · 4 620 | 155 · 8 1240 | 31 · 16 496 | 1 · 32 32 | | | 2451 |
| 6 | 1 · 1 1 | 63 · 2 126 | 651 · 4 2604 | 1395 · 8 11160 | 651 · 16 10416 | 63 · 32 2016 | 1 · 64 64 | | 26387 |
| 7 | 1 · 1 1 | 127 · 2 254 | 2667 · 4 10668 | 11811 · 8 94488 | 11811 · 16 188976 | 2667 · 32 85344 | 127 · 64 8128 | 1 · 128 128 | 387987 |
💧    triangle Sycamore    row sums Azalea :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | sums |
| 0 | 1 · 1 1 | | | | | | | | 1 |
| 1 | 0 | 1 · 2 2 | | | | | | | 2 |
| 2 | 0 | 2 · 2 4 | 1 · 4 4 | | | | | | 8 |
| 3 | 0 | 4 · 2 8 | 6 · 4 24 | 1 · 8 8 | | | | | 40 |
| 4 | 0 | 8 · 2 16 | 28 · 4 112 | 14 · 8 112 | 1 · 16 16 | | | | 256 |
| 5 | 0 | 16 · 2 32 | 120 · 4 480 | 140 · 8 1120 | 30 · 16 480 | 1 · 32 32 | | | 2144 |
| 6 | 0 | 32 · 2 64 | 496 · 4 1984 | 1240 · 8 9920 | 620 · 16 9920 | 62 · 32 1984 | 1 · 64 64 | | 23936 |
| 7 | 0 | 64 · 2 128 | 2016 · 4 8064 | 10416 · 8 83328 | 11160 · 16 178560 | 2604 · 32 83328 | 126 · 64 8064 | 1 · 128 128 | 361600 |
💧    triangle Sycamore    row sums Azalea :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | sums |
| 0 | 1 · 1 1 | | | | | | | | 1 |
| 1 | 0 | 1 · 2 2 | | | | | | | 2 |
| 2 | 0 | 1 · 4 4 | 1 · 4 4 | | | | | | 8 |
| 3 | 0 | 1 · 8 8 | 3 · 8 24 | 1 · 8 8 | | | | | 40 |
| 4 | 0 | 1 · 16 16 | 7 · 16 112 | 7 · 16 112 | 1 · 16 16 | | | | 256 |
| 5 | 0 | 1 · 32 32 | 15 · 32 480 | 35 · 32 1120 | 15 · 32 480 | 1 · 32 32 | | | 2144 |
| 6 | 0 | 1 · 64 64 | 31 · 64 1984 | 155 · 64 9920 | 155 · 64 9920 | 31 · 64 1984 | 1 · 64 64 | | 23936 |
| 7 | 0 | 1 · 128 128 | 63 · 128 8064 | 651 · 128 83328 | 1395 · 128 178560 | 651 · 128 83328 | 63 · 128 8064 | 1 · 128 128 | 361600 |
# Sequence Daisy

A006116
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 1 | 2 | 5 | 16 | 67 | 374 | 2825 | 29212 |
row sums of Oak :
🌊    triangle Oak (A022166)    row sums Daisy (A006116) :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | sums |
| 0 | 1 | | | | | | | | 1 |
| 1 | 1 | 1 | | | | | | | 2 |
| 2 | 1 | 3 | 1 | | | | | | 5 |
| 3 | 1 | 7 | 7 | 1 | | | | | 16 |
| 4 | 1 | 15 | 35 | 15 | 1 | | | | 67 |
| 5 | 1 | 31 | 155 | 155 | 31 | 1 | | | 374 |
| 6 | 1 | 63 | 651 | 1395 | 651 | 63 | 1 | | 2825 |
| 7 | 1 | 127 | 2667 | 11811 | 11811 | 2667 | 127 | 1 | 29212 |
💧    triangle Maple (~A289537)    row sums Dahlia (~A182176) :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | sums |
| 0 | 1 | | | | | | | | 1 |
| 1 | 0 | 1 | | | | | | | 1 |
| 2 | 0 | 2 | 1 | | | | | | 3 |
| 3 | 0 | 4 | 6 | 1 | | | | | 11 |
| 4 | 0 | 8 | 28 | 14 | 1 | | | | 51 |
| 5 | 0 | 16 | 120 | 140 | 30 | 1 | | | 307 |
| 6 | 0 | 32 | 496 | 1240 | 620 | 62 | 1 | | 2451 |
| 7 | 0 | 64 | 2016 | 10416 | 11160 | 2604 | 126 | 1 | 26387 |
row sums of Ash :
🌊    triangle Ash    row sums Daisy (A006116) :
| kn | 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 |
💧    triangle Aspen    row sums Dahlia (~A182176) :
| kn | 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 |

### relationships
#### Dahlia

data :
| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 🌊 Daisy | 1 | 2 | 5 | 16 | 67 | 374 | 2825 | 29212 |
| 💧 Dahlia | 1 | 1 | 3 | 11 | 51 | 307 | 2451 | 26387 |

#### Azalea

Azalea is the product of padded Daisy and the powers of two. Padded means, that there is a 1 before the other entries.
# Sequence Basil

| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 1 | 2 | 5 | 12 | 27 | 58 | 125 | 272 |

### layer sums of hyperpyramid Wisteria

c:Category:Number hyperpyramid Wisteria
row sums of Aronia :
🌊    triangle Aronia    row sums Basil :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | sums |
| 0 | 1 | | | | | | | | 1 |
| 1 | 1 | 1 | | | | | | | 2 |
| 2 | 1 | 3 | 1 | | | | | | 5 |
| 3 | 1 | 6 | 4 | 1 | | | | | 12 |
| 4 | 1 | 10 | 10 | 5 | 1 | | | | 27 |
| 5 | 1 | 15 | 20 | 15 | 6 | 1 | | | 58 |
| 6 | 1 | 21 | 36 | 37 | 22 | 7 | 1 | | 125 |
| 7 | 1 | 28 | 59 | 80 | 65 | 30 | 8 | 1 | 272 |
💧    triangle MaimedElm    row sums MaimedClover :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | sums |
| 0 | 1 | | | | | | | | 1 |
| 1 | 0 | 1 | | | | | | | 1 |
| 2 | 0 | 2 | 1 | | | | | | 3 |
| 3 | 0 | 3 | 3 | 1 | | | | | 7 |
| 4 | 0 | 4 | 6 | 4 | 1 | | | | 15 |
| 5 | 0 | 5 | 10 | 10 | 5 | 1 | | | 31 |
| 6 | 0 | 6 | 16 | 22 | 16 | 6 | 1 | | 67 |
| 7 | 0 | 7 | 23 | 43 | 43 | 23 | 7 | 1 | 147 |
row sums of Rowan :
🌊    triangle Rowan    row sums Basil :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | sums |
| 0 | 1 | | | | | | | | 1 |
| 1 | 1 | 1 | | | | | | | 2 |
| 2 | 1 | 2 | 2 | | | | | | 5 |
| 3 | 1 | 3 | 4 | 4 | | | | | 12 |
| 4 | 1 | 4 | 6 | 8 | 8 | | | | 27 |
| 5 | 1 | 5 | 8 | 12 | 16 | 16 | | | 58 |
| 6 | 1 | 6 | 10 | 16 | 24 | 32 | 36 | | 125 |
| 7 | 1 | 7 | 12 | 20 | 32 | 48 | 72 | 80 | 272 |
💧    triangle RowanDrop    row sums MaimedClover :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | sums |
| 0 | 1 | | | | | | | | 1 |
| 1 | 0 | 1 | | | | | | | 1 |
| 2 | 0 | 1 | 2 | | | | | | 3 |
| 3 | 0 | 1 | 2 | 4 | | | | | 7 |
| 4 | 0 | 1 | 2 | 4 | 8 | | | | 15 |
| 5 | 0 | 1 | 2 | 4 | 8 | 16 | | | 31 |
| 6 | 0 | 1 | 2 | 4 | 8 | 16 | 36 | | 67 |
| 7 | 0 | 1 | 2 | 4 | 8 | 16 | 36 | 80 | 147 |
row sums of Elder :
🌊    triangle Elder    row sums Basil :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | sums |
| 0 | 1 | | | | | | | | 1 |
| 1 | 1 | 1 | | | | | | | 2 |
| 2 | 2 | 2 | 1 | | | | | | 5 |
| 3 | 5 | 4 | 2 | 1 | | | | | 12 |
| 4 | 12 | 8 | 4 | 2 | 1 | | | | 27 |
| 5 | 27 | 16 | 8 | 4 | 2 | 1 | | | 58 |
| 6 | 62 | 32 | 16 | 8 | 4 | 2 | 1 | | 125 |
| 7 | 141 | 68 | 32 | 16 | 8 | 4 | 2 | 1 | 272 |
💧    triangle ElderDrop    row sums MaimedClover :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | sums |
| 0 | 1 | | | | | | | | 1 |
| 1 | 0 | 1 | | | | | | | 1 |
| 2 | 1 | 1 | 1 | | | | | | 3 |
| 3 | 3 | 2 | 1 | 1 | | | | | 7 |
| 4 | 7 | 4 | 2 | 1 | 1 | | | | 15 |
| 5 | 15 | 8 | 4 | 2 | 1 | 1 | | | 31 |
| 6 | 35 | 16 | 8 | 4 | 2 | 1 | 1 | | 67 |
| 7 | 79 | 36 | 16 | 8 | 4 | 2 | 1 | 1 | 147 |
# Sequence Lavender

| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 1 | 0 | 1 | 2 | 4 | 8 | 20 | 44 |
row sums of Magnolia :
🌊    triangle TurnedLime (~A034253)     row sums Heather (~A034343) :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | sums |
| 0 | 1 | | | | | | | | 1 |
| 1 | 1 | 0 | | | | | | | 1 |
| 2 | 1 | 1 | 0 | | | | | | 2 |
| 3 | 1 | 2 | 1 | 0 | | | | | 4 |
| 4 | 1 | 3 | 3 | 1 | 0 | | | | 8 |
| 5 | 1 | 4 | 6 | 4 | 1 | 0 | | | 16 |
| 6 | 1 | 5 | 11 | 12 | 6 | 1 | 0 | | 36 |
| 7 | 1 | 6 | 17 | 27 | 21 | 7 | 1 | 0 | 80 |
💧    triangle Magnolia    row sums Lavender :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | sums |
| 0 | 1 | | | | | | | | 1 |
| 1 | 0 | 0 | | | | | | | 0 |
| 2 | 0 | 1 | 0 | | | | | | 1 |
| 3 | 0 | 1 | 1 | 0 | | | | | 2 |
| 4 | 0 | 1 | 2 | 1 | 0 | | | | 4 |
| 5 | 0 | 1 | 3 | 3 | 1 | 0 | | | 8 |
| 6 | 0 | 1 | 5 | 8 | 5 | 1 | 0 | | 20 |
| 7 | 0 | 1 | 6 | 15 | 15 | 6 | 1 | 0 | 44 |
# Sequence MaimedClover

| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 1 | 1 | 3 | 7 | 15 | 31 | 67 | 147 |
This is Clover with all entries except the first reduced by 1.

### layer sums of hyperpyramid WisteriaDrop

c:Category:Number hyperpyramid WisteriaDrop
row sums of MaimedElm :
🌊    triangle Aronia    row sums Basil :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | sums |
| 0 | 1 | | | | | | | | 1 |
| 1 | 1 | 1 | | | | | | | 2 |
| 2 | 1 | 3 | 1 | | | | | | 5 |
| 3 | 1 | 6 | 4 | 1 | | | | | 12 |
| 4 | 1 | 10 | 10 | 5 | 1 | | | | 27 |
| 5 | 1 | 15 | 20 | 15 | 6 | 1 | | | 58 |
| 6 | 1 | 21 | 36 | 37 | 22 | 7 | 1 | | 125 |
| 7 | 1 | 28 | 59 | 80 | 65 | 30 | 8 | 1 | 272 |
💧    triangle MaimedElm    row sums MaimedClover :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | sums |
| 0 | 1 | | | | | | | | 1 |
| 1 | 0 | 1 | | | | | | | 1 |
| 2 | 0 | 2 | 1 | | | | | | 3 |
| 3 | 0 | 3 | 3 | 1 | | | | | 7 |
| 4 | 0 | 4 | 6 | 4 | 1 | | | | 15 |
| 5 | 0 | 5 | 10 | 10 | 5 | 1 | | | 31 |
| 6 | 0 | 6 | 16 | 22 | 16 | 6 | 1 | | 67 |
| 7 | 0 | 7 | 23 | 43 | 43 | 23 | 7 | 1 | 147 |
row sums of RowanDrop :
🌊    triangle Rowan    row sums Basil :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | sums |
| 0 | 1 | | | | | | | | 1 |
| 1 | 1 | 1 | | | | | | | 2 |
| 2 | 1 | 2 | 2 | | | | | | 5 |
| 3 | 1 | 3 | 4 | 4 | | | | | 12 |
| 4 | 1 | 4 | 6 | 8 | 8 | | | | 27 |
| 5 | 1 | 5 | 8 | 12 | 16 | 16 | | | 58 |
| 6 | 1 | 6 | 10 | 16 | 24 | 32 | 36 | | 125 |
| 7 | 1 | 7 | 12 | 20 | 32 | 48 | 72 | 80 | 272 |
💧    triangle RowanDrop    row sums MaimedClover :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | sums |
| 0 | 1 | | | | | | | | 1 |
| 1 | 0 | 1 | | | | | | | 1 |
| 2 | 0 | 1 | 2 | | | | | | 3 |
| 3 | 0 | 1 | 2 | 4 | | | | | 7 |
| 4 | 0 | 1 | 2 | 4 | 8 | | | | 15 |
| 5 | 0 | 1 | 2 | 4 | 8 | 16 | | | 31 |
| 6 | 0 | 1 | 2 | 4 | 8 | 16 | 36 | | 67 |
| 7 | 0 | 1 | 2 | 4 | 8 | 16 | 36 | 80 | 147 |
row sums of ElderDrop :
🌊    triangle Elder    row sums Basil :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | sums |
| 0 | 1 | | | | | | | | 1 |
| 1 | 1 | 1 | | | | | | | 2 |
| 2 | 2 | 2 | 1 | | | | | | 5 |
| 3 | 5 | 4 | 2 | 1 | | | | | 12 |
| 4 | 12 | 8 | 4 | 2 | 1 | | | | 27 |
| 5 | 27 | 16 | 8 | 4 | 2 | 1 | | | 58 |
| 6 | 62 | 32 | 16 | 8 | 4 | 2 | 1 | | 125 |
| 7 | 141 | 68 | 32 | 16 | 8 | 4 | 2 | 1 | 272 |
💧    triangle ElderDrop    row sums MaimedClover :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | sums |
| 0 | 1 | | | | | | | | 1 |
| 1 | 0 | 1 | | | | | | | 1 |
| 2 | 1 | 1 | 1 | | | | | | 3 |
| 3 | 3 | 2 | 1 | 1 | | | | | 7 |
| 4 | 7 | 4 | 2 | 1 | 1 | | | | 15 |
| 5 | 15 | 8 | 4 | 2 | 1 | 1 | | | 31 |
| 6 | 35 | 16 | 8 | 4 | 2 | 1 | 1 | | 67 |
| 7 | 79 | 36 | 16 | 8 | 4 | 2 | 1 | 1 | 147 |
# Sequence Valerian

| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 1 | 2 | 5 | 16 | 61 | 264 | 1225 | 5790 |

### layer sums of hyperpyramid Syringa

c:Category:Number hyperpyramid Syringa
row sums of Juniper :
🌊    triangle Juniper    row sums Valerian :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | sums |
| 0 | 1 | | | | | | | | 1 |
| 1 | 1 | 1 | | | | | | | 2 |
| 2 | 1 | 3 | 1 | | | | | | 5 |
| 3 | 1 | 7 | 7 | 1 | | | | | 16 |
| 4 | 1 | 15 | 29 | 15 | 1 | | | | 61 |
| 5 | 1 | 31 | 100 | 100 | 31 | 1 | | | 264 |
| 6 | 1 | 63 | 306 | 485 | 306 | 63 | 1 | | 1225 |
| 7 | 1 | 127 | 861 | 1906 | 1906 | 861 | 127 | 1 | 5790 |
💧    triangle JuniperDrop    row sums ValerianDrop :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | sums |
| 0 | 1 | | | | | | | | 1 |
| 1 | 0 | 1 | | | | | | | 1 |
| 2 | 0 | 2 | 1 | | | | | | 3 |
| 3 | 0 | 4 | 6 | 1 | | | | | 11 |
| 4 | 0 | 8 | 22 | 14 | 1 | | | | 45 |
| 5 | 0 | 16 | 71 | 85 | 30 | 1 | | | 203 |
| 6 | 0 | 32 | 206 | 385 | 275 | 62 | 1 | | 961 |
| 7 | 0 | 64 | 555 | 1421 | 1600 | 798 | 126 | 1 | 4565 |
row sums of Sloe :
🌊    triangle Sloe    row sums Valerian :
| kn | 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 · 20 20 | | | | 61 |
| 5 | 1 · 1 1 | 5 · 1 5 | 10 · 2 20 | 10 · 6 60 | 5 · 20 100 | 1 · 78 78 | | | 264 |
| 6 | 1 · 1 1 | 6 · 1 6 | 15 · 2 30 | 20 · 6 120 | 15 · 20 300 | 6 · 78 468 | 1 · 300 300 | | 1225 |
| 7 | 1 · 1 1 | 7 · 1 7 | 21 · 2 42 | 35 · 6 210 | 35 · 20 700 | 21 · 78 1638 | 7 · 300 2100 | 1 · 1092 1092 | 5790 |
💧    triangle SloeDrop    row sums ValerianDrop :
| kn | 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 · 20 20 | | | | 45 |
| 5 | 0 | 1 · 1 1 | 4 · 2 8 | 6 · 6 36 | 4 · 20 80 | 1 · 78 78 | | | 203 |
| 6 | 0 | 1 · 1 1 | 5 · 2 10 | 10 · 6 60 | 10 · 20 200 | 5 · 78 390 | 1 · 300 300 | | 961 |
| 7 | 0 | 1 · 1 1 | 6 · 2 12 | 15 · 6 90 | 20 · 20 400 | 15 · 78 1170 | 6 · 300 1800 | 1 · 1092 1092 | 4565 |
# Sequence Verbena

These sequences appear in triangle Sloe and its relatives. Triangles Sloe$_{(Drop)}$ are Pascal's triangle with columns multiplied by Verbena (so Verbena is their diagonal). Triangles TurnedSloe$_{(Drop)}$ have Verbena$_{(Drop)}$ as their left columns.
Compare sequence Aster.

## Sequences Verbena$_{(Drop)}$

| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 🌊 Verbena | 1 | 1 | 2 | 6 | 20 | 78 | 300 | 1092 |
| 💧 VerbenaDrop | 1 | 0 | 1 | 4 | 14 | 58 | 222 | 792 |

## relationships

comparison to Aster :
| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| Aster | 1 | 1 | 2 | 6 | 26 | 158 | 1330 | 15414 |
| Verbena | 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 |
| VerbenaDrop | 1 | 0 | 1 | 4 | 14 | 58 | 222 | 792 |
| Δ | 0 | 0 | 0 | 0 | 6 | 74 | 950 | 13292 |
# Sequence Aster

These sequences appear in triangle Ash and related triangles. Triangles /$_{Aspen}$ are Pascal's triangle with columns multiplied by Aster (so Aster is their diagonal). Triangles /$_{Alder}$ have Aster$_{(Drop)}$ as their left columns.
Compare sequence Verbena.

## 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.
triangle Birch :
triangle Birch (~A139382)    row sums Aster (A135922) :
| kn | 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) :
| kn | 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 :
| kn | 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 |

## relationships

comparison to Verbena :
| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| Aster | 1 | 1 | 2 | 6 | 26 | 158 | 1330 | 15414 |
| Verbena | 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 |
| VerbenaDrop | 1 | 0 | 1 | 4 | 14 | 58 | 222 | 792 |
| Δ | 0 | 0 | 0 | 0 | 6 | 74 | 950 | 13292 |