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.

A127389 Number of fixed points in range [A014137(n-1)..A014138(n-1)] of permutations A127377/A127378 and A127387.

Original entry on oeis.org

1, 1, 0, 1, 2, 4, 10, 23, 56, 138, 344, 870, 2220, 5716, 14828, 38717, 101682, 268416, 711810, 1895432, 5066030, 13586082, 36547534, 98593064, 266661162, 722953814, 1964358938, 5348367006, 14589803090, 39870312218, 109136843138
Offset: 0

Views

Author

Antti Karttunen, Jan 16 2007

Keywords

Crossrefs

This is INVERTi transform of A086625 (appropriately shifted). I.e. INVERT([1, 1, 0, 1, 2, 4, 10, 23, 56, 138, 344, 870, 2220, 5716]) gives: 1, 2, 3, 6, 12, 26, 59, 138, 332, 814, 2028, 5118, ... (beginning of A086625)

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=(1 + x*A^2)*(1+x)/(1+x+2*x^2));polcoeff(A,n)}

Formula

G.f. satisfies: A(x) = (1 + x*A(x)^2)*(1+x)/(1+x+2*x^2).

Extensions

Generating function, PARI-program and most of the terms supplied by Paul D. Hanna, Jan 15 2007

A127385 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A127387.

Original entry on oeis.org

1, 1, 1, 3, 8, 23, 71, 226, 743, 2500, 8570, 29828, 105116, 374308
Offset: 0

Views

Author

Antti Karttunen, Jan 16 2007

Keywords

Crossrefs

A127383, A127386, A127389.

A127302 Matula-Goebel signatures for plane binary trees encoded by A014486.

Original entry on oeis.org

1, 4, 14, 14, 86, 86, 49, 86, 86, 886, 886, 454, 886, 886, 301, 301, 301, 886, 886, 301, 454, 886, 886, 13766, 13766, 6418, 13766, 13766, 3986, 3986, 3986, 13766, 13766, 3986, 6418, 13766, 13766, 3101, 3101, 1589, 3101, 3101, 1849, 1849, 3101, 13766
Offset: 0

Views

Author

Antti Karttunen, Jan 16 2007

Keywords

Comments

This sequence maps A000108(n) oriented (plane) rooted binary trees encoded in range [A014137(n-1)..A014138(n-1)] of A014486 to A001190(n+1) non-oriented rooted binary trees, encoded by their Matula-Goebel numbers (when viewed as a subset of non-oriented rooted general trees). See also the comments at A127301.
If the signature-permutation of a Catalan automorphism SP satisfies the condition A127302(SP(n)) = A127302(n) for all n, then it preserves the non-oriented form of a binary tree. Examples of such automorphisms include A069770, A057163, A122351, A069767/A069768, A073286-A073289, A089854, A089859/A089863, A089864, A122282, A123492-A123494, A123715/A123716, A127377-A127380, A127387 and A127388.
A153835 divides natural numbers to same equivalence classes, i.e. a(i) = a(j) <=> A153835(i) = A153835(j) - Antti Karttunen, Jan 03 2013

Examples

			A001190(n+1) distinct values occur each range [A014137(n-1)..A014138(n-1)]. As an example, terms A014486(4..8) encode the following five plane binary trees:
........\/.....\/.................\/.....\/...
.......\/.......\/.....\/.\/.....\/.......\/..
......\/.......\/.......\_/.......\/.......\/.
n=.....4........5........6........7........8..
The trees in positions 4, 5, 7 and 8 all produce Matula-Goebel number A000040(1)*A000040(A000040(1)*A000040(A000040(1)*A000040(1))) = 2*A000040(2*A000040(2*2)) = 2*A000040(14) = 2*43 = 86, as they are just different planar representations of the one and same non-oriented tree. The tree in position 6 produces Matula-Goebel number A000040(A000040(1)*A000040(1)) * A000040(A000040(1)*A000040(1)) = A000040(2*2) * A000040(2*2) = 7*7 = 49. Thus a(4..8) = 86,86,49,86,86.

Links

Crossrefs

Cf A127301, A153835, A153829.

Formula

a(n) = A127301(A057123(n)).
Can be also computed directly as a fold, see the Scheme-program. - Antti Karttunen, Jan 03 2013

A127377 Signature-permutation of a Catalan automorphism, auxiliary bijection for Callan's 2006 bijection on Dyck Paths.

Original entry on oeis.org

0, 1, 3, 2, 8, 7, 6, 4, 5, 22, 21, 20, 17, 18, 14, 15, 19, 9, 10, 16, 11, 13, 12, 64, 63, 62, 58, 59, 54, 55, 61, 45, 46, 57, 48, 50, 49, 37, 38, 39, 41, 40, 51, 52, 60, 23, 24, 56, 25, 27, 26, 42, 43, 53, 28, 29, 44, 33, 36, 35, 47, 30, 34, 31, 32, 196, 195, 194, 189, 190
Offset: 0

Views

Author

Antti Karttunen, Jan 16 2007

Keywords

Comments

Used to construct A127379.

Links

Crossrefs

Inverse: A127378. The number of cycles and the number of fixed points in range [A014137(n-1)..A014138(n-1)] of this permutation are given by A127383 and A127389. The maximum cycles and LCM's of cycle sizes begin as 1, 1, 2, 4, 4, 8, 8, 8, 8, 16, 16, 16, 16, 16, ... A127387 shows a variant which is an involution. A127302(a(n)) = A127302(n) holds for all n.

A127388 Signature-permutation of a Catalan automorphism, a self-inverse variant of A127379.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jan 16 2007

Keywords

Comments

This automorphism is RIBS-transformation (explained in A122200) of the automorphism A127387.

Links

Crossrefs

The number of cycles and the number of fixed points in range [A014137(n-1)..A014138(n-1)] of this involution are given by A127386 and A086625 shifted once right (this automorphism has the same fixed points as A127379/A127380). A127302(a(n)) = A127302(n) holds for all n.

A153247 Number of fleeing trees computed for Catalan bijection A123493.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 0, 2, 1, 3, 2, 0, 2, 1, 3, 1, 3, 3, 2, 1, 0, 2, 1, 4, 3, 1, 3, 2, 3, 1, 3, 3, 2, 1, 0, 2, 1, 4, 3, 0, 2, 1, 2, 2, 4, 4, 3, 3, 1, 3, 2, 2, 2, 0, 3, 1, 2, 3, 3, 2, 1, 1, 0, 2, 1, 5, 4, 2, 4, 3, 4, 2, 4, 4, 3, 2, 1, 3, 2, 4, 3, 0, 2, 1, 2, 2, 4, 4, 3, 3, 1, 3, 2, 2, 2, 0, 3, 1, 2, 3, 3, 2
Offset: 0

Views

Author

Antti Karttunen, Dec 22 2008

Keywords

Comments

See comments at A153246. Essentially, A123493 does not extend uniquely to an automorphism of infinite binary tree, because its behavior is dependent on whether certain vertices of a finite binary tree are leaves (terminal nodes) or not. Similarly for bijections like A127387 and A127379.

Crossrefs

Cf. A153248.

Extensions

Edited by Charles R Greathouse IV, May 13 2010
Showing 1-6 of 6 results.

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