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.

Previous Showing 21-29 of 29 results.

A085223 Row 1 of A085201.

Original entry on oeis.org

1, 2, 4, 6, 9, 11, 14, 16, 19, 23, 25, 28, 30, 33, 37, 39, 42, 44, 47, 51, 53, 56, 60, 65, 67, 70, 72, 75, 79, 81, 84, 86, 89, 93, 95, 98, 102, 107, 109, 112, 114, 117, 121, 123, 126, 128, 131, 135, 137, 140, 144, 149, 151, 154, 156, 159, 163, 165, 168, 172, 177, 179
Offset: 0

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Author

Antti Karttunen, Jun 23 2003

Keywords

Comments

Gives in A014486 the positions of the plane general trees whose rightmost subtree (branching from the root) is just a stick: "/", thus corresponding to the parenthesizations whose last element (of the top-level list) is an empty parenthesization: (), i.e. in A063171 positions of the terms which end with digits ...10

Formula

a(n) = A085201bi(n, 1) = A057164(A072795(A057164(n))) = A057508(A072795(A057508(n))) = A080300(A085224(n))

A069769 Self-inverse permutation of natural numbers induced by the automorphism Rev1CarSide! acting on the parenthesizations encoded by A014486.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 6, 8, 9, 10, 11, 12, 13, 17, 18, 16, 14, 15, 21, 20, 19, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 45, 46, 48, 49, 50, 44, 47, 42, 37, 38, 43, 39, 40, 41, 58, 59, 57, 54, 55, 56, 53, 51, 52, 63, 62, 61, 60, 64, 65, 66, 67, 68, 69, 70, 71
Offset: 0

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Author

Antti Karttunen, Apr 16 2002

Keywords

Links

Crossrefs

The car/cdr-flipped conjugate of A057508, i.e. A069769(n) = A057163(A057508(A057163(n))). Cf. also A069787, A057161.

A073192 Number of general plane trees whose n-th subtree from the left is equal to the n-th subtree from the right, for all its subtrees (i.e., are palindromic in the shallow sense).

Original entry on oeis.org

1, 1, 2, 3, 8, 18, 54, 155, 500, 1614, 5456, 18630, 64960, 228740, 814914, 2926323, 10589916, 38561814, 141219432, 519711666, 1921142832, 7129756188, 26555149404, 99228108222, 371886574632, 1397548389644, 5265131346368
Offset: 0

Views

Author

Antti Karttunen, Jun 25 2002

Keywords

Comments

The Catalan bijection A057508 fixes only these kinds of trees, so this occurs in the table A073202 as row 168.

Links

Crossrefs

Occurs for first time in A073202 as row 168.
Cf. also A073190.

Programs

  • Maple
    A073192 := proc(n) local d; add( (`mod`((n-d+1),2))*Cat((n-d)/2)*(`if`((0=d),1,Cat(d-1))), d=0..n); end;
Cat := n -> binomial(2*n,n)/(n+1);
  • Mathematica
    a[n_] := Sum[Mod[n - k + 1, 2]*CatalanNumber[(n - k)/2]*If[k == 0, 1, CatalanNumber[k - 1]], {k, 0, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 05 2016 *)
  • PARI
    Gat(n) = if (n == -1, 1, binomial(2*n,n)/(n+1));
a(n) = sum(i=0, n, if (!((n-i)%2), Gat((n-i)/2)*Gat(i-1))); \\ Michel Marcus, May 30 2018

Formula

a(n) = Sum_{i=0..n, (n-i) is even} Gat((n-i)/2)*Gat(i-1), where Gat(-1) = 1 and otherwise like A000108(n).
A073193(n) = (A000108(n) + A073192(n))/2.

A085164 Inverse permutation to A085163.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 5, 6, 8, 9, 17, 10, 16, 21, 11, 20, 12, 14, 18, 13, 15, 19, 22, 23, 45, 24, 44, 58, 25, 54, 26, 42, 49, 27, 43, 56, 63, 28, 48, 29, 53, 62, 30, 57, 31, 37, 46, 35, 38, 47, 59, 33, 61, 34, 39, 55, 32, 40, 51, 50, 36, 41, 52, 60, 64, 65, 129, 66, 128, 170
Offset: 0

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Links

Crossrefs

Inverse: A085163. a(n) = A057508(A085161(n)). Occurs in A073200. Cf. also A085171, A085172.
Number of fixed points: A051920. See comment at A085163.

A129608 Signature-permutation of a Catalan automorphism: swap the two rightmost subtrees of general trees.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 5, 7, 8, 9, 11, 10, 16, 19, 14, 15, 12, 17, 18, 13, 20, 21, 22, 23, 25, 24, 30, 33, 28, 29, 26, 44, 47, 27, 53, 56, 60, 37, 39, 38, 43, 52, 42, 40, 31, 45, 46, 32, 48, 49, 50, 51, 41, 34, 54, 55, 35, 57, 58, 59, 36, 61, 62, 63, 64, 65, 67, 66, 72, 75, 70, 71
Offset: 0

Views

Author

Antti Karttunen, May 22 2007

Keywords

Comments

This self-inverse automorphism is obtained as either SPINE(*A129607) or ENIPS(*A129607). See the definitions given in A122203 and A122204.

Links

Crossrefs

A129608. a(n) = A057508(A072796(A057508(n))) = A057164(A072796(A057164(n))). Row 3608 of A122203 and A122204.

A130339 Signature permutation of a Catalan automorphism: swap the two rightmost subtrees of general trees, if the root degree (A057515(n)) is even.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 5, 7, 8, 9, 10, 11, 16, 19, 14, 15, 12, 17, 18, 13, 20, 21, 22, 23, 25, 24, 26, 27, 28, 29, 30, 44, 47, 33, 53, 56, 60, 37, 38, 39, 43, 52, 42, 40, 31, 45, 46, 32, 48, 49, 50, 51, 41, 34, 54, 55, 35, 57, 58, 59, 36, 61, 62, 63, 64, 65, 66, 67, 72, 75, 70, 71
Offset: 0

Views

Author

Antti Karttunen, Jun 05 2007

Keywords

Comments

This self-inverse automorphism is obtained as either SPINE(*A129608) or ENIPS(*A129608). See the definitions given in A122203 and A122204.

Links

Crossrefs

Cf. a(n) = A057508(A130340(A057508(n))) = A057164(A130340(A057164(n))). Row 3608 of A122285 and A122286. a(n) = A129608(n), if A057515(n) mod 2 = 0, otherwise a(n)=n.

A130340 Signature permutation of a Catalan automorphism: swap the two leftmost subtrees of general trees, if the root degree (A057515(n)) is even.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 5, 7, 8, 9, 10, 11, 16, 19, 14, 15, 12, 17, 18, 13, 20, 21, 22, 23, 24, 25, 26, 27, 37, 29, 30, 44, 47, 33, 53, 56, 60, 28, 38, 39, 43, 52, 42, 40, 31, 45, 46, 32, 48, 49, 50, 51, 41, 34, 54, 55, 35, 57, 58, 59, 36, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71
Offset: 0

Views

Author

Antti Karttunen, Jun 05 2007

Keywords

Comments

This is a self-inverse automorfism (an involution). Can be used to construct A130373.

Links

Crossrefs

Cf. a(n) = A057508(A130339(A057508(n))) = A057164(A130339(A057164(n))). a(n) = A072796(n), if A057515(n) mod 2 = 0, otherwise a(n)=n.

A130373 Signature permutation of a Catalan automorphism: flip the positions of even- and odd-indexed elements at the top level of the list, leaving the first element in place if the length (A057515(n)) is odd.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 5, 7, 8, 9, 11, 10, 16, 19, 14, 15, 12, 17, 18, 13, 20, 21, 22, 23, 25, 24, 30, 33, 37, 29, 26, 44, 47, 27, 53, 56, 60, 28, 39, 38, 43, 52, 42, 40, 31, 45, 46, 32, 48, 49, 50, 51, 41, 34, 54, 55, 35, 57, 58, 59, 36, 61, 62, 63, 64, 65, 67, 66, 72, 75, 79, 71
Offset: 0

Views

Author

Antti Karttunen, Jun 05 2007

Keywords

Comments

This self-inverse automorphism permutes the top level of a list of even length (1 2 3 4 ... 2n-1 2n) as (2 1 4 3 ... 2n 2n-1), and when applied to a list of odd length (1 2 3 4 5 ... 2n 2n+1), permutes it as (1 3 2 5 4 ... 2n+1 2n).

Links

Crossrefs

Cf. A057508, A057164, A057164.
SPINE and ENIPS transform of *A130340 (transformations explained in A122203 and A122204).
The number of cycles and the number of fixed points in range [A014137(n-1)..A014138(n-1)] of this permutation are given by A073193 and A073192.

Formula

a(n) = A057508(A130374(A057508(n))) = A057164(A130374(A057164(n))).

A130374 Signature permutation of a Catalan automorphism: flip the positions of even- and odd-indexed elements at the top level of the list, leaving the last element in place if the length (A057515(n)) is odd.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 5, 7, 8, 9, 10, 14, 16, 19, 11, 15, 12, 17, 18, 13, 20, 21, 22, 23, 25, 24, 26, 27, 37, 38, 42, 44, 47, 51, 53, 56, 60, 28, 29, 39, 43, 52, 30, 40, 31, 45, 46, 32, 48, 49, 50, 33, 41, 34, 54, 55, 35, 57, 58, 59, 36, 61, 62, 63, 64, 65, 66, 70, 72, 75, 67, 71
Offset: 0

Views

Author

Antti Karttunen, Jun 05 2007

Keywords

Comments

This self-inverse automorphism permutes the top level of a list of even length (1 2 3 4 ... 2n-1 2n) as (2 1 4 3 ... 2n 2n-1), and when applied to a list of odd length (1 2 3 4 ... 2n-1 2n 2n+1), permutes it as (2 1 4 3 ... 2n 2n-1 2n+1).

Links

Crossrefs

Cf. a(n) = A057508(A130373(A057508(n))) = A057164(A130373(A057164(n))) = A127285(A127288(n)) = A127287(A127286(n)). Also a(A085223(n)) = A130370(A122282(A130369(A085223(n)))) holds for all n>=0. The number of cycles and the number of fixed points in range [A014137(n-1)..A014138(n-1)] of this permutation are given by A073193 and A073192.
Previous Showing 21-29 of 29 results.

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