arity: top to bottom depth: back to front valency: left to right
Indices in the image go from 1 to 7. Liana is always 1 where depth and valency are 0. But this column is not shown in the images.
The sum ignoring valency is triangle Oak. The sum ignoring depth is triangle Ash.
The layer sums (and row sums of these triangles) are sequence Daisy (A006116).
fixed arity (depth × valency matrices)
The row sums are rows of triangle Oak. The column sums are rows of triangle Ash. The total sums are entries of Daisy.
arity 0
v
d
0
1
2
3
4
5
6
7
Σ
0
1
1
1
2
3
4
5
6
7
Σ
1
1
arity 1
v
d
0
1
2
3
4
5
6
7
Σ
0
1
1
1
1
1
2
3
4
5
6
7
Σ
1
1
2
arity 2
v
d
0
1
2
3
4
5
6
7
Σ
0
1
1
1
2
1
3
2
1
1
3
4
5
6
7
Σ
1
2
2
5
arity 3
v
d
0
1
2
3
4
5
6
7
Σ
0
1
1
1
3
3
1
7
2
3
4
7
3
1
1
4
5
6
7
Σ
1
3
6
6
16
arity 4
v
d
0
1
2
3
4
5
6
7
Σ
0
1
1
1
4
6
4
1
15
2
6
16
13
35
3
4
11
15
4
1
1
5
6
7
Σ
1
4
12
24
26
67
arity 5
v
d
0
1
2
3
4
5
6
7
Σ
0
1
1
1
5
10
10
5
1
31
2
10
40
65
40
155
3
10
55
90
155
4
5
26
31
5
1
1
6
7
Σ
1
5
20
60
130
158
374
arity 6
v
d
0
1
2
3
4
5
6
7
Σ
0
1
1
1
6
15
20
15
6
1
63
2
15
80
195
240
121
651
3
20
165
540
670
1395
4
15
156
480
651
5
6
57
63
6
1
1
7
Σ
1
6
30
120
390
948
1330
2825
arity 7
v
d
0
1
2
3
4
5
6
7
Σ
0
1
1
1
7
21
35
35
21
7
1
127
2
21
140
455
840
847
364
2667
3
35
385
1890
4690
4811
11811
4
35
546
3360
7870
11811
5
21
399
2247
2667
6
7
120
127
7
1
1
Σ
1
7
42
210
910
3318
9310
15414
29212
fixed depth (arity × valency matrices)
The row sums are columns of triangle Oak.
depth 0
v
a
0
1
2
3
4
5
6
7
Σ
0
1
1
1
1
1
2
1
1
3
1
1
4
1
1
5
1
1
6
1
1
7
1
1
depth 1
v
a
0
1
2
3
4
5
6
7
Σ
0
1
1
1
2
2
1
3
3
3
3
1
7
4
4
6
4
1
15
5
5
10
10
5
1
31
6
6
15
20
15
6
1
63
7
7
21
35
35
21
7
1
127
depth 2
v
a
0
1
2
3
4
5
6
7
Σ
0
1
2
1
1
3
3
4
7
4
6
16
13
35
5
10
40
65
40
155
6
15
80
195
240
121
651
7
21
140
455
840
847
364
2667
depth 3
v
a
0
1
2
3
4
5
6
7
Σ
0
1
2
3
1
1
4
4
11
15
5
10
55
90
155
6
20
165
540
670
1395
7
35
385
1890
4690
4811
11811
depth 4
v
a
0
1
2
3
4
5
6
7
Σ
0
1
2
3
4
1
1
5
5
26
31
6
15
156
480
651
7
35
546
3360
7870
11811
depth 5
v
a
0
1
2
3
4
5
6
7
Σ
0
1
2
3
4
5
1
1
6
6
57
63
7
21
399
2247
2667
depth 6
v
a
0
1
2
3
4
5
6
7
Σ
0
1
2
3
4
5
6
1
1
7
7
120
127
depth 7
v
a
0
1
2
3
4
5
6
7
Σ
0
1
2
3
4
5
6
7
1
1
sum: triangle Ash
v
a
0
1
2
3
4
5
6
7
Σ
0
1
1
1
1
1
2
2
1
2
2
5
3
1
3
6
6
16
4
1
4
12
24
26
67
5
1
5
20
60
130
158
374
6
1
6
30
120
390
948
1330
2825
7
1
7
42
210
910
3318
9310
15414
29212
fixed valency (arity × depth matrices)
The row sums are columns of triangle Ash.
valency 0
d
a
0
1
2
3
4
5
6
7
Σ
0
1
1
1
1
1
2
1
1
3
1
1
4
1
1
5
1
1
6
1
1
7
1
1
valency 1
d
a
0
1
2
3
4
5
6
7
Σ
0
1
1
1
2
2
2
3
3
3
4
4
4
5
5
5
6
6
6
7
7
7
valency 2
d
a
0
1
2
3
4
5
6
7
Σ
0
1
2
1
1
2
3
3
3
6
4
6
6
12
5
10
10
20
6
15
15
30
7
21
21
42
valency 3
d
a
0
1
2
3
4
5
6
7
Σ
0
1
2
3
1
4
1
6
4
4
16
4
24
5
10
40
10
60
6
20
80
20
120
7
35
140
35
210
valency 4
d
a
0
1
2
3
4
5
6
7
Σ
0
1
2
3
4
1
13
11
1
26
5
5
65
55
5
130
6
15
195
165
15
390
7
35
455
385
35
910
valency 5
d
a
0
1
2
3
4
5
6
7
Σ
0
1
2
3
4
5
1
40
90
26
1
158
6
6
240
540
156
6
948
7
21
840
1890
546
21
3318
valency 6
d
a
0
1
2
3
4
5
6
7
Σ
0
1
2
3
4
5
6
1
121
670
480
57
1
1330
7
7
847
4690
3360
399
7
9310
valency 7
d
a
0
1
2
3
4
5
6
7
Σ
0
1
2
3
4
5
6
7
1
364
4811
7870
2247
120
1
15414
sum: triangle Oak
d
a
0
1
2
3
4
5
6
7
Σ
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
other sides
valency − depth = 0 Pascal's triangle (with left column not shown)
adicity: top to bottom depth: back to front valency: left to right
Indices in the image go from 1 to 7.
The entry Ivy(0, 0, 0) = 1 is not shown in the images.
The sum ignoring valency is triangle Maple. The sum ignoring depth is triangle Aspen.
The layer sums (and row sums of these triangles) are sequence Dahlia (A182176).
The pyramid sides in the back (depth = 1) and front (valency − depth = 0) are Pascal's triangle.
fixed adicity (depth × valency matrices)
The row sums are rows of triangle Maple. The column sums are rows of triangle Aspen. The total sums are entries of Dahlia.
adicity 0
v
d
0
1
2
3
4
5
6
7
Σ
0
1
1
1
2
3
4
5
6
7
Σ
1
1
adicity 1
v
d
0
1
2
3
4
5
6
7
Σ
0
1
1
1
2
3
4
5
6
7
Σ
1
1
adicity 2
v
d
0
1
2
3
4
5
6
7
Σ
0
1
1
1
2
2
1
1
3
4
5
6
7
Σ
1
2
3
adicity 3
v
d
0
1
2
3
4
5
6
7
Σ
0
1
1
2
1
4
2
2
4
6
3
1
1
4
5
6
7
Σ
1
4
6
11
adicity 4
v
d
0
1
2
3
4
5
6
7
Σ
0
1
1
3
3
1
8
2
3
12
13
28
3
3
11
14
4
1
1
5
6
7
Σ
1
6
18
26
51
adicity 5
v
d
0
1
2
3
4
5
6
7
Σ
0
1
1
4
6
4
1
16
2
4
24
52
40
120
3
6
44
90
140
4
4
26
30
5
1
1
6
7
Σ
1
8
36
104
158
307
adicity 6
v
d
0
1
2
3
4
5
6
7
Σ
0
1
1
5
10
10
5
1
32
2
5
40
130
200
121
496
3
10
110
450
670
1240
4
10
130
480
620
5
5
57
62
6
1
1
7
Σ
1
10
60
260
790
1330
2451
adicity 7
v
d
0
1
2
3
4
5
6
7
Σ
0
1
1
6
15
20
15
6
1
64
2
6
60
260
600
726
364
2016
3
15
220
1350
4020
4811
10416
4
20
390
2880
7870
11160
5
15
342
2247
2604
6
6
120
126
7
1
1
Σ
1
12
90
520
2370
7980
15414
26387
fixed depth (adicity × valency matrices)
The row sums are columns of triangle Maple.
depth 0
v
a
0
1
2
3
4
5
6
7
Σ
0
1
1
1
2
3
4
5
6
7
depth 1
v
a
0
1
2
3
4
5
6
7
Σ
0
1
1
1
2
1
1
2
3
1
2
1
4
4
1
3
3
1
8
5
1
4
6
4
1
16
6
1
5
10
10
5
1
32
7
1
6
15
20
15
6
1
64
depth 2
v
a
0
1
2
3
4
5
6
7
Σ
0
1
2
1
1
3
2
4
6
4
3
12
13
28
5
4
24
52
40
120
6
5
40
130
200
121
496
7
6
60
260
600
726
364
2016
depth 3
v
a
0
1
2
3
4
5
6
7
Σ
0
1
2
3
1
1
4
3
11
14
5
6
44
90
140
6
10
110
450
670
1240
7
15
220
1350
4020
4811
10416
depth 4
v
a
0
1
2
3
4
5
6
7
Σ
0
1
2
3
4
1
1
5
4
26
30
6
10
130
480
620
7
20
390
2880
7870
11160
depth 5
v
a
0
1
2
3
4
5
6
7
Σ
0
1
2
3
4
5
1
1
6
5
57
62
7
15
342
2247
2604
depth 6
v
a
0
1
2
3
4
5
6
7
Σ
0
1
2
3
4
5
6
1
1
7
6
120
126
depth 7
v
a
0
1
2
3
4
5
6
7
Σ
0
1
2
3
4
5
6
7
1
1
sum: triangle Aspen
v
a
0
1
2
3
4
5
6
7
Σ
0
1
1
1
1
1
2
1
2
3
3
1
4
6
11
4
1
6
18
26
51
5
1
8
36
104
158
307
6
1
10
60
260
790
1330
2451
7
1
12
90
520
2370
7980
15414
26387
fixed valency (adicity × depth matrices)
The row sums are columns of triangle Aspen.
valency 0
d
a
0
1
2
3
4
5
6
7
Σ
0
1
1
1
2
3
4
5
6
7
valency 1
d
a
0
1
2
3
4
5
6
7
Σ
0
1
1
1
2
1
1
3
1
1
4
1
1
5
1
1
6
1
1
7
1
1
valency 2
d
a
0
1
2
3
4
5
6
7
Σ
0
1
2
1
1
2
3
2
2
4
4
3
3
6
5
4
4
8
6
5
5
10
7
6
6
12
valency 3
d
a
0
1
2
3
4
5
6
7
Σ
0
1
2
3
1
4
1
6
4
3
12
3
18
5
6
24
6
36
6
10
40
10
60
7
15
60
15
90
valency 4
d
a
0
1
2
3
4
5
6
7
Σ
0
1
2
3
4
1
13
11
1
26
5
4
52
44
4
104
6
10
130
110
10
260
7
20
260
220
20
520
valency 5
d
a
0
1
2
3
4
5
6
7
Σ
0
1
2
3
4
5
1
40
90
26
1
158
6
5
200
450
130
5
790
7
15
600
1350
390
15
2370
valency 6
d
a
0
1
2
3
4
5
6
7
Σ
0
1
2
3
4
5
6
1
121
670
480
57
1
1330
7
6
726
4020
2880
342
6
7980
valency 7
d
a
0
1
2
3
4
5
6
7
Σ
0
1
2
3
4
5
6
7
1
364
4811
7870
2247
120
1
15414
sum: triangle Maple
d
a
0
1
2
3
4
5
6
7
Σ
0
1
1
1
1
1
2
2
1
3
3
4
6
1
11
4
8
28
14
1
51
5
16
120
140
30
1
307
6
32
496
1240
620
62
1
2451
7
64
2016
10416
11160
2604
126
1
26387
other sides
valency − depth = 0 Pascal's triangle (trivial column on the left not shown)
