A080300 -id:A080300 - cp's OEIS frontend

A085223 Row 1 of A085201.

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

Antti Karttunen, Jun 23 2003

nonn

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

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

A071163 A014486-indices for rooted binary trees with height equal to number of internal vertices. (Binary trees where at each internal vertex at least the other child is leaf.)

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 12, 13, 17, 18, 21, 22, 23, 24, 26, 27, 31, 32, 35, 36, 45, 46, 49, 50, 58, 59, 63, 64, 65, 66, 68, 69, 73, 74, 77, 78, 87, 88, 91, 92, 100, 101, 105, 106, 129, 130, 133, 134, 142, 143, 147, 148, 170, 171, 175, 176, 189, 190, 195, 196, 197

Offset: 0

Antti Karttunen, May 14 2002

nonn

This subset of integers is closed by the actions of A069770, A057163, A069767, A069768, A122353, A122354, A122301, A122302, etc. (meaning, e.g., that A069767(a(n)) is a member from this sequence for all n), that is, by any Catalan bijection which is an image of some element of the automorphism group of infinite binary tree (the latter in a sense given by Grigorchuk, et al., being isomorphic to an infinitely iterated wreath product of cyclic groups of two elements). See the comments about the isomorphism "psi" given at A153141.

a(n) could be probably computed directly from the binary expansion of n by using a (somewhat) similar ranking function as given in A209640, but utilizing A009766 instead of A007318.

Antti Karttunen, Table of n, a(n) for n = 0..65535

a(n) = A080300(A071162(n)).

A080116 Characteristic function of A014486. a(n) = 1 if n's binary expansion is totally balanced, otherwise zero.

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Antti Karttunen, Feb 11 2003

a(n) = 1 if the binary representation of n forms a valid Dyck path, or equally, a well-formed parenthesization when 1's are converted to left and 0's to right parentheses (that is, when A007088(n) is in A063171), and 0 otherwise. - Antti Karttunen, Aug 23 2019

			0 stands for an empty parenthesization, thus a(0) = 1.
2 has binary expansion "10", which corresponds with "()", thus a(2) = 1.
3 has binary expansion "11", but "((" is not a well-formed parenthesization, thus a(3) = 0.
10 has binary expansion "1010", corresponding with a well-formed parenthesization "()()", thus a(10) = 1.
38 has binary expansion "100110", but "())(()" is not a well-formed parenthesization, thus a(38) = 0.

Cf. A014486, A063171, A080110, A080111, A080300, A080301.

A080116 := proc(n) local c,lev; lev := 0; c := n; while(c > 0) do lev := lev + (-1)^c; c := floor(c/2); if(lev < 0) then RETURN(0); fi; od; if(lev > 0) then RETURN(0); else RETURN(1); fi; end;

A080116[n_] := (lev = 0; c = n; While[c > 0, lev = lev + (-1)^c; c = Floor[c/2]; If[lev < 0, Return[0]]]; If[lev > 0, Return[0], Return[1]]); Table[A080116[n], {n, 0, 104}] (* Jean-François Alcover, Jul 24 2013, translated from Maple *)

A080116(n) = { my(k=0); while(n, k += (-1)^n; n >>= 1; if(k<0, return(0))); (0==k); }; \\ Antti Karttunen, Aug 23 2019

def A080116(n) :
    lev = 0
    while n > 0 :
        lev += (-1)^n
        if lev < 0: return 0
        n = n//2
    return 0 if lev > 0 else 1
[A080116(n) for n in (0..104)] # Peter Luschny, Aug 09 2012

Examples added by Antti Karttunen, Aug 23 2019

A080295 A014486-index of the branch-reduced binomial-mod-2 binary trees.

Original entry on oeis.org

1, 6, 477, 70818, 2208159610, 445557105328, 19185898282602827, 905428099582719818595

Offset: 0

Antti Karttunen, Mar 02 2003

nonn

Cf. A014486, A080265, A080293, A080296, A080300, A080320, A080979.

a(n) = A080298(A080265(n)).
a(n) = A080979(A080265(2*n)).
a(n) = A080300(A080293(n)).

A218777 A014486-indices for the Beanstalk-tree growing one natural number at time, starting from the tree of one internal node (1), with the "lesser numbers to the left hand side" construction.

Original entry on oeis.org

1, 3, 6, 15, 39, 117, 359, 1149, 3780, 12752, 43740, 152346, 536655, 1909411, 6852153, 24772328, 90137210, 329838392, 1213048022, 4481262944, 16621500330, 61875433940, 231099456721, 865736972546, 3252139562356, 12247622197965, 46232602224402, 174897265479074

Offset: 1

Antti Karttunen, Nov 17 2012

nonn

See comments and examples at A218776.

Antti Karttunen, Table of n, a(n) for n = 1..128

a(n) = A057163(A218779(n)). Cf. A218787, A218776, A218781, A218783.

a(n) = A080300(A218776(n)).

A218779 A014486-indices for the Beanstalk-tree growing one natural number at time, starting from the tree of one internal node (1), with the "lesser numbers to the right hand side" construction.

Original entry on oeis.org

1, 2, 6, 16, 53, 156, 491, 1584, 5286, 17855, 61405, 214012, 754731, 2688307, 9659023, 34947356, 127260566, 465920651, 1714294995, 6335423798, 23506865260, 87534068725, 327024845325, 1225395355165, 4604242172697, 17343284212635, 65480496260109, 247755060623522

Offset: 1

Antti Karttunen, Nov 17 2012

nonn

See comments and examples at A218778.

Antti Karttunen, Table of n, a(n) for n = 1..128

Cf. A218778, A218781, A218783.

a(n) = A080300(A218778(n)).

a(n) = A057163(A218777(n)).

A218781 A014486-indices for the compact representation of Beanstalk-tree, with the lesser numbers coming to the left hand side.

Original entry on oeis.org

1, 2, 5, 12, 32, 92, 278, 877, 2861, 9572, 32656, 113164, 397190, 1409006, 5043617, 18194197, 66075777, 241385044, 886422017, 3270283189, 12115355601, 45052126049, 168100964161, 629171367473, 2361546968519, 8886942571534, 33523357596518, 126736969302857

Offset: 1

Antti Karttunen, Nov 17 2012

nonn

See the comments and examples at A218780.

Antti Karttunen, Table of n, a(n) for n = 1..128

Cf. A218787, A218780, A218777, A218779.

a(n) = A080300(A218780(n)).

A082858 Array A(x,y): the greatest common subtree (intersect) of the binary trees x and y, (x,y) running as (0,0),(1,0),(0,1),(2,0),(1,1),(0,2) and each index referring to a binary tree encoded by A014486(j).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 2, 3, 2, 1, 0, 0, 1, 2, 1, 1, 2, 1, 0, 0, 1, 2, 1, 4, 1, 2, 1, 0, 0, 1, 1, 3, 2, 2, 3, 1, 1, 0, 0, 1, 1, 3, 2, 5, 2, 3, 1, 1, 0, 0, 1, 2, 3, 1, 2, 2, 1, 3, 2, 1, 0, 0, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 1, 0, 0, 1, 2, 1, 4, 1, 3, 3, 1, 4, 1, 2, 1, 0, 0, 1, 2, 1, 4, 2, 3, 7, 3, 2, 4, 1, 2, 1, 0

Offset: 0

Antti Karttunen, May 06 2003

Note that together with A082860 this forms a distributive lattice, thus it is possible to compute this function also with the binary AND-operation (A004198) with the help of appropriate mapping functions. I.e. we have A(x,y) = A082857(A004198(A082856(x), A082856(y))).

Antti Karttunen, Alternative Catalan Orderings (with the complete Scheme source)
Index entries for sequences related to lattices

Cf. A072764. The lower/upper triangular region: A082859. Cf. A080300, A025581, A002262.

A084108 A014486-indices of "Complete Binary Trees".

Original entry on oeis.org

0, 1, 6, 477, 11231586, 17656351387745509, 118547604486270210927391203275078974, 14557702344245589436016960628730576845591277100880695377777962217288601549

Offset: 0

Antti Karttunen, May 13 2003

nonn

Fixed points of permutations A069767 and A069768.

a(n) = A057117(A083942(n)). Also iterates of A080298, i.e., a(1)=A080298(0), a(2)=A080298(A080298(0)), a(3)=A080298(A080298(A080298(0))), etc. Cf. also A083940, A080274.

a(n) = A080300(A084107(n)).

A127300 Signature-permutation of A057164-conjugate of the inverse of Elizalde's and Deutsch's 2003 bijection for Dyck paths.

Original entry on oeis.org

0, 1, 3, 2, 8, 4, 7, 6, 5, 22, 11, 17, 16, 12, 21, 9, 20, 19, 10, 18, 14, 13, 15, 64, 33, 48, 47, 34, 58, 23, 57, 56, 24, 49, 37, 35, 38, 63, 30, 45, 44, 31, 62, 28, 61, 60, 29, 46, 42, 32, 43, 59, 25, 54, 53, 26, 50, 39, 36, 40, 55, 51, 27, 52, 41, 196, 102, 145, 144, 103

Offset: 0

Antti Karttunen, Jan 16 2007

nonn

Used to construct the inverse for A127291.

Emeric Deutsch and Sergi Elizalde, A simple and unusual bijection for Dyck paths and its consequences, Annals of Combinatorics, 7 (2003), no. 3, 281-297.

Index entries for signature-permutations of Catalan automorphisms

Inverse: A127299. a(n) = A057164(A127292(A057164(n))) = A127290(A057164(n)). Cf. A014486.

cp's OEIS Frontend

A085223 Row 1 of A085201.

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A071163 A014486-indices for rooted binary trees with height equal to number of internal vertices. (Binary trees where at each internal vertex at least the other child is leaf.)

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A080116 Characteristic function of A014486. a(n) = 1 if n's binary expansion is totally balanced, otherwise zero.

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A080295 A014486-index of the branch-reduced binomial-mod-2 binary trees.

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A218777 A014486-indices for the Beanstalk-tree growing one natural number at time, starting from the tree of one internal node (1), with the "lesser numbers to the left hand side" construction.

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A218779 A014486-indices for the Beanstalk-tree growing one natural number at time, starting from the tree of one internal node (1), with the "lesser numbers to the right hand side" construction.

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A218781 A014486-indices for the compact representation of Beanstalk-tree, with the lesser numbers coming to the left hand side.

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Formula

A082858 Array A(x,y): the greatest common subtree (intersect) of the binary trees x and y, (x,y) running as (0,0),(1,0),(0,1),(2,0),(1,1),(0,2) and each index referring to a binary tree encoded by A014486(j).

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A084108 A014486-indices of "Complete Binary Trees".

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A127300 Signature-permutation of A057164-conjugate of the inverse of Elizalde's and Deutsch's 2003 bijection for Dyck paths.

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