Template:Seal pyramids Wisteria and Lonicera

🌊    pyramid Wisteria

overview arity: top to bottom depth: back to front cohort weight: left to right The sum ignoring cohort weight is triangle Oak. The sum ignoring depth is reflected Ash. The layer sums (and row sums of these triangles) are sequence Daisy (A006116). fixed arity    (depth × cohort weight matrices) The row sums are rows of Oak. The column sums are rows of reflected Ash. The total sums are entries of Daisy. arity 0 c 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 c 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 c d 0 1 2 3 4 5 6 7 Σ 0 1 1 1 1 2 3 2 1 1 3 4 5 6 7 Σ 2 2 1 5 arity 3 c d 0 1 2 3 4 5 6 7 Σ 0 1 1 1 4 3 7 2 1 3 3 7 3 1 1 4 5 6 7 Σ 6 6 3 1 16 arity 4 c d 0 1 2 3 4 5 6 7 Σ 0 1 1 1 11 4 15 2 13 16 6 35 3 1 4 6 4 15 4 1 1 5 6 7 Σ 26 24 12 4 1 67 arity 5 c d 0 1 2 3 4 5 6 7 Σ 0 1 1 1 26 5 31 2 90 55 10 155 3 40 65 40 10 155 4 1 5 10 10 5 31 5 1 1 6 7 Σ 158 130 60 20 5 1 374 arity 6 c d 0 1 2 3 4 5 6 7 Σ 0 1 1 1 57 6 63 2 480 156 15 651 3 670 540 165 20 1395 4 121 240 195 80 15 651 5 1 6 15 20 15 6 63 6 1 1 7 Σ 1330 948 390 120 30 6 1 2825 arity 7 c d 0 1 2 3 4 5 6 7 Σ 0 1 1 1 120 7 127 2 2247 399 21 2667 3 7870 3360 546 35 11811 4 4811 4690 1890 385 35 11811 5 364 847 840 455 140 21 2667 6 1 7 21 35 35 21 7 127 7 1 1 Σ 15414 9310 3318 910 210 42 7 1 29212 fixed depth    (arity × cohort weight matrices) The row sums are columns of Oak. depth 0 c 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 The left column shows the Euler numbers (A000295). Their positive terms are summations of 1, 3, 7, 15, 31... c a 0 1 2 3 4 5 6 7 Σ 0 1 1 1 2 1 2 3 3 4 3 7 4 11 4 15 5 26 5 31 6 57 6 63 7 120 7 127 depth 2 c a 0 1 2 3 4 5 6 7 Σ 0 1 2 1 1 3 1 3 3 7 4 13 16 6 35 5 90 55 10 155 6 480 156 15 651 7 2247 399 21 2667 depth 3 c a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 1 1 4 1 4 6 4 15 5 40 65 40 10 155 6 670 540 165 20 1395 7 7870 3360 546 35 11811 depth 4 c a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 1 1 5 1 5 10 10 5 31 6 121 240 195 80 15 651 7 4811 4690 1890 385 35 11811 depth 5 c a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 5 1 1 6 1 6 15 20 15 6 63 7 364 847 840 455 140 21 2667 depth 6 c a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 5 6 1 1 7 1 7 21 35 35 21 7 127 depth 7 c a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 5 6 7 1 1 sum:   reflected triangle Ash c a 0 1 2 3 4 5 6 7 Σ 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 fixed cohort weight    (arity × depth matrices) The row sums are columns of reflected Ash. cohort weight 0 d a 0 1 2 3 4 5 6 7 Σ 0 1 1 1 1 1 2 1 1 2 3 1 4 1 6 4 1 11 13 1 26 5 1 26 90 40 1 158 6 1 57 480 670 121 1 1330 7 1 120 2247 7870 4811 364 1 15414 cohort weight 1 d a 0 1 2 3 4 5 6 7 Σ 0 1 1 1 2 2 2 3 3 3 6 4 4 16 4 24 5 5 55 65 5 130 6 6 156 540 240 6 948 7 7 399 3360 4690 847 7 9310 cohort weight 2 d a 0 1 2 3 4 5 6 7 Σ 0 1 2 1 1 3 3 3 4 6 6 12 5 10 40 10 60 6 15 165 195 15 390 7 21 546 1890 840 21 3318 cohort weight 3 d a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 1 1 4 4 4 5 10 10 20 6 20 80 20 120 7 35 385 455 35 910 cohort weight 4 d a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 1 1 5 5 5 6 15 15 30 7 35 140 35 210 cohort weight 5 d a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 5 1 1 6 6 6 7 21 21 42 cohort weight 6 d a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 5 6 1 1 7 7 7 cohort weight 7 d a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 5 6 7 1 1 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 depth − cohort weight = 0 Pascal's triangle adicity − depth = 1 almost Pascal's triangle   (without right diagonal)

💧    pyramid Lonicera

overview adicity: top to bottom depth: back to front cohort weight: left to right The sum ignoring cohort weight is triangle Maple. The sum ignoring depth is triangle Alder. The layer sums (and row sums of these triangles) are sequence Dahlia (A182176). fixed adicity    (depth × cohort weight matrices) The row sums are rows of Maple. The column sums are rows of Alder. The total sums are entries of Dahlia. adicity 0 c 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 c 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 c 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 1 1 3 adicity 3 c d 0 1 2 3 4 5 6 7 Σ 0 1 3 1 4 2 1 3 2 6 3 1 1 4 5 6 7 Σ 4 4 2 1 11 adicity 4 c d 0 1 2 3 4 5 6 7 Σ 0 1 7 1 8 2 12 13 3 28 3 1 4 6 3 14 4 1 1 5 6 7 Σ 20 18 9 3 1 51 adicity 5 c d 0 1 2 3 4 5 6 7 Σ 0 1 15 1 16 2 77 39 4 120 3 39 61 34 6 140 4 1 5 10 10 4 30 5 1 1 6 7 Σ 132 106 48 16 4 1 307 adicity 6 c d 0 1 2 3 4 5 6 7 Σ 0 1 31 1 32 2 390 101 5 496 3 630 475 125 10 1240 4 120 235 185 70 10 620 5 1 6 15 20 15 5 62 6 1 1 7 Σ 1172 818 330 100 25 5 1 2451 adicity 7 c d 0 1 2 3 4 5 6 7 Σ 0 1 63 1 64 2 1767 243 6 2016 3 7200 2820 381 15 10416 4 4690 4450 1695 305 20 11160 5 363 841 825 435 125 15 2604 6 1 7 21 35 35 21 6 126 7 1 1 Σ 14084 8362 2928 790 180 36 6 1 26387 fixed depth    (adicity × cohort weight matrices) The row sums are columns of Maple. depth 0 c a 0 1 2 3 4 5 6 7 Σ 0 1 1 1 2 3 4 5 6 7 depth 1 c a 0 1 2 3 4 5 6 7 Σ 0 1 1 1 2 1 1 2 3 3 1 4 4 7 1 8 5 15 1 16 6 31 1 32 7 63 1 64 depth 2 c a 0 1 2 3 4 5 6 7 Σ 0 1 2 1 1 3 1 3 2 6 4 12 13 3 28 5 77 39 4 120 6 390 101 5 496 7 1767 243 6 2016 depth 3 c a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 1 1 4 1 4 6 3 14 5 39 61 34 6 140 6 630 475 125 10 1240 7 7200 2820 381 15 10416 depth 4 c a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 1 1 5 1 5 10 10 4 30 6 120 235 185 70 10 620 7 4690 4450 1695 305 20 11160 depth 5 c a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 5 1 1 6 1 6 15 20 15 5 62 7 363 841 825 435 125 15 2604 depth 6 c a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 5 6 1 1 7 1 7 21 35 35 21 6 126 depth 7 c a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 5 6 7 1 1 sum:   triangle Alder c a 0 1 2 3 4 5 6 7 Σ 0 1 1 1 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 fixed cohort weight    (adicity × depth matrices) The row sums are columns of Alder. cohort weight 0 d a 0 1 2 3 4 5 6 7 Σ 0 1 1 1 2 1 1 3 3 1 4 4 7 12 1 20 5 15 77 39 1 132 6 31 390 630 120 1 1172 7 63 1767 7200 4690 363 1 14084 cohort weight 1 d a 0 1 2 3 4 5 6 7 Σ 0 1 1 1 2 1 1 3 1 3 4 4 1 13 4 18 5 1 39 61 5 106 6 1 101 475 235 6 818 7 1 243 2820 4450 841 7 8362 cohort weight 2 d a 0 1 2 3 4 5 6 7 Σ 0 1 2 1 1 3 2 2 4 3 6 9 5 4 34 10 48 6 5 125 185 15 330 7 6 381 1695 825 21 2928 cohort weight 3 d a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 1 1 4 3 3 5 6 10 16 6 10 70 20 100 7 15 305 435 35 790 cohort weight 4 d a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 1 1 5 4 4 6 10 15 25 7 20 125 35 180 cohort weight 5 d a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 5 1 1 6 5 5 7 15 21 36 cohort weight 6 d a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 5 6 1 1 7 6 6 cohort weight 7 d a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 5 6 7 1 1 File:Number pyramid Lonicera; cohort weight 7.png 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 depth − cohort weight = 0 Pascal's triangle   (with trivial column on the left) adicity − depth = 1 almost Pascal's triangle   (without right diagonal, next reduced by 1)

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Seal (discrete mathematics)#Quantities