Template:Seal pyramids Liana and Ivy

🌊    pyramid Liana

overview 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 along valency is triangle Oak. The sum along depth is triangle Ash. The layer sums (and row sums of these triangles) are sequence Daisy. 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) arity − valency = 0 triangle Birch

💧    pyramid Ivy

overview Indices in the image go from 1 to 7. The entry Ivy(0, 0, 0) = 1 is not shown in the images. The sum along valency is triangle [[|Maple]]. The sum along depth is triangle Aspen. The layer sums (and row sums of these triangles) are sequence Dahlia. 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) adicity − valency = 0 triangle Birch

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Seal (discrete mathematics)#Quantities