A202694 -id:A202694 - cp's OEIS frontend

A202695 The first of a set of three triangles constructed by the same rule as A202692-A202694, but where the top entries in the three triangles are 1,0,0 respectively.

1, 0, 1, 5, 5, 4, 0, 5, 10, 14, 205, 205, 200, 190, 176

Offset: 1

N. J. A. Sloane, Dec 23 2011

			The triangle begins:
1
0 1
5 5 4
0 5 10 14
205 205 200 190 176
...
The three triangles together:
1                      0                      0
0 1                    1 1                    1 1
5 5 4                  4 3 2                  2 1 0
0 5 10 14              14 18 21 23            23 25 26 26
205 205 200 190 176    176 162 144 123 100    100 77 52 26 0

M. Josuat-Verges, J.-C. Novelli and J.-Y. Thibon, The algebraic combinatorics of snakes, arXiv preprint arXiv:1110.5272, 2011

The three triangles are A202695, A202696, A202697.

A202698 The first of a set of three triangles constructed by the same rule as A202692-A202694, but where the top entries in the three triangles are 0,1,0 respectively.

Original entry on oeis.org

0, 0, 0, 3, 3, 3, 0, 3, 6, 9, 147, 147, 144, 138, 129

Offset: 1

N. J. A. Sloane, Dec 23 2011

			The triangle begins:
0
0 0
3 3 3
0 3 6 9
147 147 144 138 129
...
The three triangles together:
0                   1                        0
0 0                 0 1                      1 1
3 3 3               3 3 2                    2 1 0
0 3 6 9             9 12 15 17               17 19 20 20
147 147 144 138 129 129 120 108 93 76        76 59 40 20 0

M. Josuat-Verges, J.-C. Novelli and J.-Y. Thibon, The algebraic combinatorics of snakes, arXiv preprint arXiv:1110.5272, 2011

The three triangles are A202698, A202699, A202700.

A202701 The first of a set of three triangles constructed by the same rule as A202692-A202694, but where the top entries in the three triangles are 0,0,1 respectively.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 0, 1, 2, 3, 53, 53, 52, 50, 47

Offset: 1

N. J. A. Sloane, Dec 23 2011

			The triangle begins:
0
0 0
1 1 1
0 1 2 3
53 53 52 50 47
...
The three triangles together:
0                 0                  1
0 0               0 0                0 1
1 1 1             1 1 1              1 1 0
0 1 2 3           3 4 5 6            6 7 8 8
53 53 52 50 47    47 44 40 35 29     29 23 16 8 0

M. Josuat-Verges, J.-C. Novelli and J.-Y. Thibon, The algebraic combinatorics of snakes, arXiv preprint arXiv:1110.5272, 2011

The three triangles are A202701, A202702, A202703.

A202696 The second of a set of three triangles constructed by the same rule as A202692-A202694, but where the top entries in the three triangles are 1,0,0 respectively.

Original entry on oeis.org

0, 1, 1, 4, 3, 2, 14, 18, 21, 23, 176, 162, 144, 123, 100

Offset: 1

N. J. A. Sloane, Dec 23 2011

			Triangle begins:
0
1 1
4 3 2
14 18 21 23
176 162 144 123 100
...

M. Josuat-Verges, J.-C. Novelli and J.-Y. Thibon, The algebraic combinatorics of snakes, arXiv preprint arXiv:1110.5272, 2011

See A202695 for further information.

A202697 The third of a set of three triangles constructed by the same rule as A202692-A202694, but where the top entries in the three triangles are 1,0,0 respectively.

Original entry on oeis.org

0, 1, 1, 2, 1, 0, 23, 25, 26, 26, 100, 77, 52, 26, 0

Offset: 1

N. J. A. Sloane, Dec 23 2011

			Triangle begins:
0
1 1
2 1 0
23 25 26 26
100 77 52 26 0
...

M. Josuat-Verges, J.-C. Novelli and J.-Y. Thibon, The algebraic combinatorics of snakes, arXiv preprint arXiv:1110.5272, 2011

See A202695 for further information.

A202699 The second of a set of three triangles constructed by the same rule as A202692-A202694, but where the top entries in the three triangles are 0,1,0 respectively.

Original entry on oeis.org

1, 0, 1, 3, 3, 2, 9, 12, 15, 17, 129, 120, 108, 93, 76

Offset: 1

N. J. A. Sloane, Dec 23 2011

			Triangle begins:
1
0 1
3 3 2
9 12 15 17
129 120 108 93 76
...

M. Josuat-Verges, J.-C. Novelli and J.-Y. Thibon, The algebraic combinatorics of snakes, arXiv preprint arXiv:1110.5272, 2011

See A202698 for further information.

A202700 The third of a set of three triangles constructed by the same rule as A202692-A202694, but where the top entries in the three triangles are 0,1,0 respectively.

Original entry on oeis.org

0, 1, 1, 2, 1, 0, 17, 19, 20, 20, 76, 59, 40, 20, 0

Offset: 1

N. J. A. Sloane, Dec 23 2011

			Triangle begins:
0
1 1
2 1 0
17 19 20 20
76 59 40 20 0
...

M. Josuat-Verges, J.-C. Novelli and J.-Y. Thibon, The algebraic combinatorics of snakes, arXiv preprint arXiv:1110.5272, 2011

See A202698 for further information.

A202702 The second of a set of three triangles constructed by the same rule as A202692-A202694, but where the top entries in the three triangles are 0,0,1 respectively.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 3, 4, 5, 6, 47, 44, 40, 35, 29

Offset: 1

N. J. A. Sloane, Dec 23 2011

			Triangle begins:
0
0 0
1 1 1
3 4 5 6
47 44 40 35 29
...

M. Josuat-Verges, J.-C. Novelli and J.-Y. Thibon, The algebraic combinatorics of snakes, arXiv preprint arXiv:1110.5272, 2011

See A202701 for further information.

A202703 The third of a set of three triangles constructed by the same rule as A202692-A202694, but where the top entries in the three triangles are 0,0,1 respectively.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 6, 7, 8, 8, 29, 23, 16, 8, 0

Offset: 1

N. J. A. Sloane, Dec 23 2011

			Triangle begins:
1
0 1
1 1 0
6 7 8 8
29 23 16 8 0
...

M. Josuat-Verges, J.-C. Novelli and J.-Y. Thibon, The algebraic combinatorics of snakes, arXiv preprint arXiv:1110.5272, 2011

See A202701 for further information.

A202692 The first of three triangles counting 3-colored alternating permutations by their last value.

Original entry on oeis.org

1, 0, 1, 9, 9, 8, 0, 9, 18, 26, 405, 405, 396, 378, 352, 0, 405, 810, 1206, 1584, 1936, 44469, 44469, 44064, 43254, 42048, 40464, 38528, 0, 44469, 88938, 133002, 176256, 218304, 258768, 297296, 9086985, 9086985, 9042516, 8953578, 8820576, 8644320, 8426016, 8167248, 7869952

Offset: 1

N. J. A. Sloane, Dec 23 2011

"The table counting alternating permutations with r colors by their last value is obtained by the following algorithm: first separate the picture by the column p = 0 and then compute r triangles. Put 1 at the top of each triangle and compute the rest as follows: fill the second row of all triangles as the sum of the elements of the first row strictly to their left. Then fill the third row of all triangles as the sum of the elements of the previous row to their right. Compute all rows successively by reading from left to right and right to left alternately." [Joshuat-Verges et al.]

			The triangle begins:
1
0 1
9 9 8
0 9 18 26
405 405 396 378 352
...
The three triangles together:
1                     1                      1
0 1                   1 2                    2 3
9 9 8                 8 7 5                  5 3 0
0 9 18 26             26 34 41 46            46 51 54 54
405 405 396 378 352   352 326 292 251 205    205 159 108 54 0

Georg Fischer and Sean. A. Irvine, Java program source code
M. Josuat-Verges, J.-C. Novelli and J.-Y. Thibon, The algebraic combinatorics of snakes, arXiv preprint arXiv:1110.5272 [math.CO], 2011.

The three triangles with initial 1,1,1 are A202692, A202693, A202694; with 1,0,0: A202695, A202696, A202697; with 0,1,0: A202698, A202699, A202700; with 0,0,1: A202701, A202702, A202703.

Java
```
/* Cf. link */
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More terms from Georg Fischer, Oct 20 2021

cp's OEIS Frontend

A202695 The first of a set of three triangles constructed by the same rule as A202692-A202694, but where the top entries in the three triangles are 1,0,0 respectively.

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A202698 The first of a set of three triangles constructed by the same rule as A202692-A202694, but where the top entries in the three triangles are 0,1,0 respectively.

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A202701 The first of a set of three triangles constructed by the same rule as A202692-A202694, but where the top entries in the three triangles are 0,0,1 respectively.

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A202696 The second of a set of three triangles constructed by the same rule as A202692-A202694, but where the top entries in the three triangles are 1,0,0 respectively.

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A202697 The third of a set of three triangles constructed by the same rule as A202692-A202694, but where the top entries in the three triangles are 1,0,0 respectively.

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A202699 The second of a set of three triangles constructed by the same rule as A202692-A202694, but where the top entries in the three triangles are 0,1,0 respectively.

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A202700 The third of a set of three triangles constructed by the same rule as A202692-A202694, but where the top entries in the three triangles are 0,1,0 respectively.

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A202702 The second of a set of three triangles constructed by the same rule as A202692-A202694, but where the top entries in the three triangles are 0,0,1 respectively.

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A202703 The third of a set of three triangles constructed by the same rule as A202692-A202694, but where the top entries in the three triangles are 0,0,1 respectively.

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A202692 The first of three triangles counting 3-colored alternating permutations by their last value.

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