cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A307696 Number of evolutionary duplication-loss-histories with n leaves of the caterpillar species tree with 2 leaves.

Original entry on oeis.org

2, 7, 34, 200, 1318, 9354, 69864, 541323, 4310950, 35066384, 290081932, 2432766082, 20635672664, 176727482860, 1526000459400, 13270616752680, 116124930068670, 1021736927603190, 9033726534916920, 80220639767921370, 715166816624282820, 6398357633173869600
Offset: 1

Views

Author

Michael Wallner, Apr 22 2019

Keywords

Comments

An evolutionary history of size n is an ordered rooted (incomplete) binary tree with n leaves describing the evolution of a gene family of a species in phylogenomics. The caterpillar species tree S of size k is a binary tree with k leaves, where any left child is a leaf. Any node of the history is associated to a unique node of S, where specifically every leaf is associated to a leaf of S. A history is created by the following process (note that intermediate trees in this process may not be valid histories): Start with a root node associated to the root of S. For a given tree in the growth process, choose a leaf and perform a duplication, speciation, or (speciation-)loss event. A duplication event creates two children both associated to the same node as its parent. A speciation or (speciation-)loss event can only occur if the node is associated to an internal node in S. In that case, a speciation event creates two children associated to the children of the node in S. A (speciation-)loss event creates only a left or right child, associated to the left or right child in S, respectively.

Examples

			The caterpillar species tree with 2 leaves is equal to
    a
   / \
  1   2
For convenience the internal node is labeled by a, and the leaves by 1,2. The associated nodes in the histories will be denoted by the same labels.
The a(1)=2 histories with n=1 leaf are created by the following growth process:
    a       a
   /         \
  1           2
after one loss event each.
The a(2)=7 histories with n=2 leaves are created by the following growth process:
    a         a     a            a         a         a         a
   / \       /       \          / \       / \       / \       / \
  1   2     1         2        a   a     a   a     a   a     a   a
           / \       / \      /   /     /     \     \   \    \   /
          1   1     2   2    1   1     1       2     2   2    2 1

Links

Crossrefs

Caterpillar species tree sequences: A000108 (1 leaf), A307697 (3 leaves), A307698 (4 leaves), A307700 (5 leaves).

Programs

  • PARI
    my(z='z+O('z^30)); Vec(1/2-(1/2)*sqrt(-5+6*sqrt(1-4*z)+4*z)) \\ Michel Marcus, Apr 22 2019

Formula

G.f.: 1/2 - (1/2)*sqrt(-5 + 6*sqrt(1-4*z) + 4*z).

A307698 Number of evolutionary duplication-loss-histories with n leaves of the caterpillar species tree with 4 leaves.

Original entry on oeis.org

4, 39, 495, 7235, 115303, 1948791, 34379505, 626684162, 11722058693, 223870302588, 4349161774626, 85701267415112, 1709101664822416, 34432888701965454, 699810795294490974, 14331183304458656628, 295434131968070459359, 6125911207605272841753, 127680054133385458855845
Offset: 1

Views

Author

Michael Wallner, Apr 22 2019

Keywords

Comments

An evolutionary history of size n is an ordered rooted (incomplete) binary tree with n leaves describing the evolution of a gene family of a species in phylogenomics. The caterpillar species tree S of size k is a binary tree with k leaves, where any left child is a leaf. Any node of the history is associated to a unique node of S, where specifically every leaf is associated to a leaf of S. A history is created by the following process (note that intermediate trees in this process may not be valid histories): Start with a root node associated to the root of S. For a given tree in the growth process, choose a leaf and perform a duplication, speciation, or (speciation-)loss event. A duplication event creates two children both associated to the same node as its parent. A speciation or (speciation-)loss event can only occur if the node is associated to an internal node in S. In that case, a speciation event creates two children associated to the children of the node in S. A (speciation-)loss event creates only a left or right child, associated to the left or right child in S, respectively.

Examples

			The caterpillar species tree with 4 leaves is equal to
        a
       / \
      b   4
     / \
    c   3
   / \
  1   2
For convenience the internal nodes are labeled by a,b,c, and the leaves by 1,2,3,4. The associated nodes in the histories will be denoted by the same labels.
The a(1)=4 histories with n=1 leaf are created by the following growth process:
        a     a     a    a
       /     /     /      \
      b     b     b        4
     /     /       \
    c     c         3
   /       \
  1         2
after three loss events each.

Links

Crossrefs

Caterpillar species tree sequences: A000108 (1 leaf), A307696 (2 leaves), A307697 (3 leaves), A307700 (5 leaves).

Programs

  • PARI
    my(z = 'z + O('z^25), t = sqrt(1-4*z), u = sqrt(-5+6*t+4*z), v = sqrt(-t*u+3*t+3*u-4)); Vec(1/2-(1/2)*sqrt(-4-t*v+3*t+3*v)) \\ Michel Marcus, May 07 2019

Formula

G.f.: 1/2 - (1/2)*sqrt(-4 - t*v + 3*t + 3*v) where t = sqrt(1 - 4*z), u = sqrt(-5 + 6*t + 4*z) and v = sqrt(-t*u + 3*t + 3*u - 4).

A307700 Number of evolutionary duplication-loss-histories with n leaves of the caterpillar species tree with 5 leaves.

Original entry on oeis.org

5, 69, 1230, 24843, 541315, 12426996, 296546600, 7292489761, 183702242491, 4719659859582, 123261298705663, 3263950145153931, 87452457544863592, 2366980142343757033, 64628573978046899555, 1778185743733577832862, 49254755849062502247446, 1372455474283175885070422
Offset: 1

Views

Author

Michael Wallner, Apr 22 2019

Keywords

Comments

An evolutionary history of size n is an ordered rooted (incomplete) binary tree with n leaves describing the evolution of a gene family of a species in phylogenomics. The caterpillar species tree S of size k is a binary tree with k leaves, where any left child is a leaf. Any node of the history is associated to a unique node of S, where specifically every leaf is associated to a leaf of S. A history is created by the following process (note that intermediate trees in this process may not be valid histories): Start with a root node associated to the root of S. For a given tree in the growth process, choose a leaf and perform a duplication, speciation, or (speciation-)loss event. A duplication event creates two children both associated to the same node as its parent. A speciation or (speciation-)loss event can only occur if the node is associated to an internal node in S. In that case, a speciation event creates two children associated to the children of the node in S. A (speciation-)loss event creates only a left or right child, associated to the left or right child in S, respectively.

Examples

			The caterpillar species tree with 5 leaves is equal to
          a
         / \
        b   5
       / \
      c   4
     / \
    d   3
   / \
  1   2
For convenience the internal nodes are labeled by a,b,c,d, and the leaves by 1,2,3,4,5. The associated nodes in the histories will be denoted by the same labels.
The a(1)=5 histories with n=1 leaf are created by the following growth process:
          a     a     a     a    a
         /     /     /     /      \
        b     b     b     b        5
       /     /     /       \
      c     c     c         4
     /     /       \
    d     d         3
   /       \
  1         2
after four loss events each.

Links

Crossrefs

Caterpillar species tree sequences: A000108 (1 leaf), A307696 (2 leaves), A307697 (3 leaves), A307698 (4 leaves).

Programs

  • PARI
    my(z = 'z + O('z^25), t = sqrt(1-4*z), u = sqrt(-5+6*t+4*z), v = sqrt(-t*u+3*t+3*u-4), w = sqrt(-t*v+3*t+3*v-4)); Vec(1/2-(1/2)*sqrt(-4-t*w+3*t+3*w)) \\ Michel Marcus, May 07 2019

Formula

G.f.: 1/2 - (1/2)*sqrt(-4 - t*w + 3*t + 3*w) where t = sqrt(1 - 4*z), u = sqrt(-5 + 6*t + 4*z), v = sqrt(-t*u + 3*t + 3*u - 4) and w = sqrt(-t*v + 3*t + 3*v - 4).

A307941 Number of evolutionary duplication-loss-histories of the complete binary species tree with 4 leaves.

Original entry on oeis.org

4, 34, 368, 4685, 66416, 1013268, 16279788, 271594611, 4660794200, 81747301898, 1458812278424, 26400987754054, 483374731032868, 8936983620559660, 166617056922535080, 3128790129161470470, 59124052722375912960, 1123458655726125274620, 21452847767668402271220
Offset: 1

Views

Author

Cedric Chauve, May 07 2019

Keywords

Comments

An evolutionary history of size n is an ordered rooted (incomplete) binary tree with n leaves describing the evolution of a gene family of a species in phylogenomics. The complete binary species tree S of size k is a complete binary tree with k leaves. Any node of the history is associated to a unique node of S, where specifically every leaf is associated to a leaf of S. A history is created by the following process (note that intermediate trees in this process may not be valid histories): Start with a root node associated to the root of S. For a given tree in the growth process, choose a leaf and perform a duplication, speciation, or (speciation-)loss event. A duplication event creates two children both associated to the same node as its parent. A speciation or (speciation-)loss event can only occur if the node is associated to an internal node in S. In that case, a speciation event creates two children associated to the children of the node in S. A (speciation-)loss event creates only a left or right child, associated to the left or right child in S, respectively.

Examples

			The complete binary species tree with 4 leaves is equal to
     a
   /   \
  b     c
/ \   / \
1   2  3  4
For convenience the internal nodes are labeled by a,b,c and the leaves by 1,2,3,4. The associated nodes in the histories will be denoted by the same labels.
The a(1)=4 histories with n=1 leaf are created by the following growth process:
    a     a     a      a
   /     /       \      \
  b     b         c      c
/       \       /        \
1         2     3          4
after two loss events each.

Crossrefs

Cf. A000108 (caterpillar/complete binary species tree with 1 leaf, ordinary binary trees).
Cf. A307696, A307697, A307698, A307700 (caterpillar species tree with 2, 3, 4, 5 leaves).

Programs

  • PARI
    z='z+O('z^20); Vec(1/2-(1/2)*sqrt(1+6*sqrt(-5+6*sqrt(1-4*z)+4*z)-6*sqrt(1-4*z)-4*z)) \\ Jianing Song, Jul 29 2019

Formula

G.f.: 1/2-(1/2)*sqrt(1+6*sqrt(-5+6*sqrt(1-4*z)+4*z)-6*sqrt(1-4*z)-4*z).

A307943 Number of evolutionary duplication-loss-histories of the complete binary species tree with 16 leaves.

Original entry on oeis.org

16, 616, 28832, 1556780, 93017264, 5971377672, 403667945712, 28346017000314, 2048467088599520, 151362953286590792, 11383212160213595696, 868385902978402717696, 67032303753464250574432, 5225869642113491897295040, 410865063418648682500317120
Offset: 1

Views

Author

Cedric Chauve, May 07 2019

Keywords

Comments

An evolutionary history of size n is an ordered rooted (incomplete) binary tree with n leaves describing the evolution of a gene family of a species in phylogenomics. The complete binary species tree S of size k is a complete binary tree with k leaves. Any node of the history is associated to a unique node of S, where specifically every leaf is associated to a leaf of S. A history is created by the following process (note that intermediate trees in this process may not be valid histories): Start with a root node associated to the root of S. For a given tree in the growth process, choose a leaf and perform a duplication, speciation, or (speciation-)loss event. A duplication event creates two children both associated to the same node as its parent. A speciation or (speciation-)loss event can only occur if the node is associated to an internal node in S. In that case, a speciation event creates two children associated to the children of the node in S. A (speciation-)loss event creates only a left or right child, associated to the left or right child in S, respectively.

Examples

			See A307941 (complete binary species tree with 4 leaves).

Links

Crossrefs

Cf. A000108 (caterpillar/complete binary species tree with 1 leaf, ordinary binary trees).
Cf. A307696, A307697, A307698, A307700 (caterpillar species tree with 2, 3, 4, 5 leaves).

Formula

G.f.: 1/2-(1/2)*sqrt(1-6*v+6*w+6*u-6*t-4*z) where t = sqrt(1-4*z), u = sqrt(-5+6*t+4*z), v = sqrt(1+6*u-6*t-4*z) and w = sqrt(-5+6*v-6*u+6*t+4*z)

A307942 Number of evolutionary duplication-loss-histories of the complete binary species tree with 8 leaves.

Original entry on oeis.org

8, 148, 3376, 89390, 2624872, 82866636, 2755019736, 95135709027, 3380416782760, 122798718575216, 4539685792433848, 170225552910292438, 6458330316575589176, 247456381334355675220, 9561546562984390785960, 372141845574597078971490, 14575950501012888889866120
Offset: 1

Views

Author

Cedric Chauve, May 07 2019

Keywords

Comments

An evolutionary history of size n is an ordered rooted (incomplete) binary tree with n leaves describing the evolution of a gene family of a species in phylogenomics. The complete binary species tree S of size k is a complete binary tree with k leaves. Any node of the history is associated to a unique node of S, where specifically every leaf is associated to a leaf of S. A history is created by the following process (note that intermediate trees in this process may not be valid histories): Start with a root node associated to the root of S. For a given tree in the growth process, choose a leaf and perform a duplication, speciation, or (speciation-)loss event. A duplication event creates two children both associated to the same node as its parent. A speciation or (speciation-)loss event can only occur if the node is associated to an internal node in S. In that case, a speciation event creates two children associated to the children of the node in S. A (speciation-)loss event creates only a left or right child, associated to the left or right child in S, respectively.

Examples

			See A307941 (complete binary species tree with 4 leaves).

Crossrefs

Cf. A000108 (caterpillar/complete binary species tree with 1 leaf, ordinary binary trees).
Cf. A307696, A307697, A307698, A307700 (caterpillar species tree with 2, 3, 4, 5 leaves).

Programs

  • PARI
    z='z+O('z^20); my(t = sqrt(1-4*z), u = sqrt(-5+6*t+4*z), v = sqrt(1+6*u-6*t-4*z)); Vec(1/2-(1/2)*sqrt(-5+6*v-6*u+6*t+4*z)) \\ Jianing Song, Jul 29 2019

Formula

G.f.: 1/2-(1/2)*sqrt(-5+6*v-6*u+6*t+4*z) where t = sqrt(1-4*z), u = sqrt(-5+6*t+4*z) and v = sqrt(1+6*u-6*t-4*z).
Showing 1-6 of 6 results.

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