A288952 -id:A288952 - cp's OEIS frontend

A288950 Number of relaxed compacted binary trees of right height at most one with empty initial and final sequence on level 0.

1, 0, 1, 2, 15, 140, 1575, 20790, 315315, 5405400, 103378275, 2182430250, 50414138775, 1264936572900, 34258698849375, 996137551158750, 30951416768146875, 1023460181133390000, 35885072600989486875, 1329858572860198631250, 51938365373373313209375

Offset: 0

Michael Wallner, Jun 20 2017

nonn

A relaxed compacted binary tree of size n is a directed acyclic graph consisting of a binary tree with n internal nodes, one leaf, and n pointers. It is constructed from a binary tree of size n, where the first leaf in a post-order traversal is kept and all other leaves are replaced by pointers. These links may point to any node that has already been visited by the post-order traversal. The right height is the maximal number of right-edges (or right children) on all paths from the root to any leaf after deleting all pointers. The number of unbounded relaxed compacted binary trees of size n is A082161(n). The number of relaxed compacted binary trees of right height at most one of size n is A001147(n). See the Genitrini et al. and Wallner link. - Michael Wallner, Apr 20 2017

a(n) is the number of plane increasing trees with n+1 nodes where node 3 is at depth 1 on the right of node 2 and where the node n+1 has a left sibling. See the Wallner link. - Michael Wallner, Apr 20 2017

			Denote by L the leaf and by o nodes. Every node has exactly two out-going edges or pointers. Internal edges are denoted by - or |. Pointers are omitted and may point to any node further right. The root is at level 0 at the very left.
The general structure is
  L-o-o-o-o-o-o-o-o-o
    |       |     | |
    o   o-o-o   o-o o.
For n=0 the a(0)=1 solution is L.
For n=1 we have a(1)=0 because we need nodes on level 0 and level 1.
For n=2 the a(2)=1 solution is
     L-o
       |
       o
and the pointers of the node on level 1 both point to the leaf.
For n=3 the a(3)=2 solutions have the structure
     L-o
       |
     o-o
where the pointers of the last node have to point to the leaf, but the pointer of the next node has 2 choices: the leaf of the previous node.

Antoine Genitrini, Bernhard Gittenberger, Manuel Kauers and Michael Wallner, Asymptotic Enumeration of Compacted Binary Trees, arXiv:1703.10031 [math.CO], 2017.
Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017.

Cf. A001147 (relaxed compacted binary trees of right height at most one).
Cf. A082161 (relaxed compacted binary trees of unbounded right height).
Cf. A000032, A000246, A001879, A051577, A177145, A213527, A288950, A288952, A288953, A288954 (subclasses of relaxed compacted binary trees of right height at most one, see the Wallner link).
Cf. A000166, A000255, A000262, A052852, A123023, A130905, A176408, A201203 (variants of relaxed compacted binary trees of right height at most one, see the Wallner link).
Cf. A001879.

terms = 21; (z + (1 - z)/3*(2 - z + (1 - 2z)^(-1/2)) + O[z]^terms // CoefficientList[#, z] &) Range[0, terms-1]! (* Jean-François Alcover, Dec 04 2018 *)

E.g.f.: z + (1-z)/3 * (2-z + (1-2*z)^(-1/2)).
From Seiichi Manyama, Apr 26 2025: (Start)
a(n) = (n-1)*(2*n-3)/(n-2) * a(n-1) for n > 3.
a(n) = A001879(n-2)/3 for n > 2. (End)

A288953 Number of relaxed compacted binary trees of right height at most one with minimal sequences between branch nodes except after the last branch node on level 0.

Original entry on oeis.org

1, 1, 3, 10, 51, 280, 1995, 15120, 138075, 1330560, 14812875, 172972800, 2271359475, 31135104000, 471038042475, 7410154752000, 126906349444875, 2252687044608000, 43078308695296875, 851515702861824000, 17984171447178811875, 391697223316439040000

Offset: 0

Michael Wallner, Jun 20 2017

nonn

A relaxed compacted binary tree of size n is a directed acyclic graph consisting of a binary tree with n internal nodes, one leaf, and n pointers. It is constructed from a binary tree of size n, where the first leaf in a post-order traversal is kept and all other leaves are replaced by pointers. These links may point to any node that has already been visited by the post-order traversal. The right height is the maximal number of right-edges (or right children) on all paths from the root to any leaf after deleting all pointers. A branch node is a node with a left and right edge (no pointer). See the Genitrini et al. link. - Michael Wallner, Apr 20 2017

a(n) is the number of plane increasing trees with n+1 nodes where in the growth process induced by the labels maximal young leaves and non-maximal young leaves alternate except for a sequence of maximal young leaves at the beginning. A young leaf is a leaf with no left sibling. A maximal young leaf is a young leaf with maximal label. See the Wallner link. - Michael Wallner, Apr 20 2017

			Denote by L the leaf and by o nodes. Every node has exactly two out-going edges or pointers. Internal edges are denoted by - or |. Pointers are omitted and may point to any node further right. The root is at level 0 at the very left.
The general structure is
  L-o-o-o-o-o-o-o-o
          | | | | |
          o o o o o.
For n=0 the a(0)=1 solution is L.
For n=1 the a(1)=1 solution is L-o.
For n=2 the a(2)=3 solutions are
L-o-o     L-o
            |
            o
  2    +   1    solutions of this shape with pointers.

Antoine Genitrini, Bernhard Gittenberger, Manuel Kauers and Michael Wallner, Asymptotic Enumeration of Compacted Binary Trees, arXiv:1703.10031 [math.CO], 2017
Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017

Cf. A288954 (variation with additional initial sequence).
Cf. A177145 (variation without final sequence).
Cf. A001147 (relaxed compacted binary trees of right height at most one).
Cf. A082161 (relaxed compacted binary trees of unbounded right height).
Cf. A000032, A000246, A001879, A051577, A213527, A288950, A288952, A288954 (subclasses of relaxed compacted binary trees of right height at most one, see the Wallner link).
Cf. A000166, A000255, A000262, A052852, A123023, A130905, A176408, A201203 (variants of relaxed compacted binary trees of right height at most one, see the Wallner link).

E.g.f.: (2-z)/(3*(1-z)^2) + 1/(3*sqrt(1-z^2)).

A288954 Number of relaxed compacted binary trees of right height at most one with minimal sequences between branch nodes except before the first and after the last branch node on level 0.

Original entry on oeis.org

1, 1, 3, 13, 79, 555, 4605, 42315, 436275, 4894155, 60125625, 794437875, 11325612375, 172141044075, 2793834368325, 48009995908875, 874143494098875, 16757439016192875, 338309837281040625, 7157757510792763875, 158706419654857449375, 3673441093896736036875

Offset: 2

Michael Wallner, Jun 20 2017

nonn

A relaxed compacted binary tree of size n is a directed acyclic graph consisting of a binary tree with n internal nodes, one leaf, and n pointers. It is constructed from a binary tree of size n, where the first leaf in a post-order traversal is kept and all other leaves are replaced by pointers. These links may point to any node that has already been visited by the post-order traversal. The right height is the maximal number of right-edges (or right children) on all paths from the root to any leaf after deleting all pointers. A branch node is a node with a left and right edge (no pointer). See the Genitrini et al. link. - Michael Wallner, Apr 20 2017

a(n) is the number of plane increasing trees with n+1 nodes where in the growth process induced by the labels a maximal young leaves and non-maximal young leaves alternate except for a sequence of maximal young leaves at the begininning and at the end. A young leaf is a leaf with no left sibling. A maximal young leaf is a young leaf with maximal label. See the Wallner link. - Michael Wallner, Apr 20 2017

			See A288950 and A288953.

Antoine Genitrini, Bernhard Gittenberger, Manuel Kauers and Michael Wallner, Asymptotic Enumeration of Compacted Binary Trees, arXiv:1703.10031 [math.CO], 2017.
Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017.

Cf. A288953 (variation without initial sequence).
Cf. A177145 (variation without initial and final sequence).
Cf. A001147 (relaxed compacted binary trees of right height at most one).
Cf. A082161 (relaxed compacted binary trees of unbounded right height).
Cf. A000032, A000246, A001879, A051577, A213527, A288950, A288952 (subclasses of relaxed compacted binary trees of right height at most one, see the Wallner link).
Cf. A000166, A000255, A000262, A052852, A123023, A130905, A176408, A201203 (variants of relaxed compacted binary trees of right height at most one, see the Wallner link).

terms = 22; egf = 1/(3(1-z))(1/Sqrt[1-z^2] + (3z^3 - z^2 - 2z + 2)/((1-z)(1-z^2))) + O[z]^terms;
CoefficientList[egf, z] Range[0, terms-1]! (* Jean-François Alcover, Dec 13 2018 *)

E.g.f.: 1/(3*(1-z))*( 1/sqrt(1-z^2) + (3*z^3-z^2-2*z+2)/((1-z)*(1-z^2)) ).

A316666 Number of simple relaxed compacted binary trees of right height at most one with no sequences on level 1 and no final sequences on level 0.

Original entry on oeis.org

1, 0, 1, 3, 15, 87, 597, 4701, 41787, 413691, 4512993, 53779833, 695000919, 9680369943, 144560191149, 2303928046437, 39031251610227, 700394126116851, 13270625547477177, 264748979672169681, 5547121478845459983, 121784530649198053263, 2795749225338111831429, 66981491857058929294653

Offset: 0

Michael Wallner, Jul 10 2018

nonn

A relaxed compacted binary tree of size n is a directed acyclic graph consisting of a binary tree with n internal nodes, one leaf, and at most n pointers. It is constructed from a binary tree of size n, where the first leaf in a post-order traversal is kept and all other leaves are replaced by pointers. These links may point to any node that has already been visited by the post-order traversal. It is called simple if for nodes with two pointers both point to the same node. The right height is the maximal number of right-edges (or right children) on all paths from the root to any leaf after deleting all pointers. See the Wallner link.

a(n) is one of two "basis" sequences for sequences of the form a(0)=a, a(1)=b, a(n) = n*a(n-1) + (n-1)*a(n-2), the second basis sequence being A096654 (with 0 appended as a(0)). The sum of these sequences is listed as A000255. - Gary Detlefs, Dec 11 2018

G. C. Greubel, Table of n, a(n) for n = 0..448
Antoine Genitrini, Bernhard Gittenberger, Manuel Kauers and Michael Wallner, Asymptotic Enumeration of Compacted Binary Trees, arXiv:1703.10031 [math.CO], 2017.
Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017.

Cf. A000032, A000246, A001879, A051577, A213527, A288950, A288952, A288953 (subclasses of relaxed compacted binary trees of right height at most one, see the Wallner link).
Cf. A000166, A000255, A000262, A052852, A123023, A130905, A176408, A201203 (variants of simple relaxed compacted binary trees of right height at most one, see the Wallner link).
Cf. A000255, A096654.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (3*Exp(-x) + x-2)/(1-x)^2 )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Dec 12 2018

aseq := n-> 3*round((n+2)*n!/exp(1))-(n+2)*n!: bseq := n-> (n+2)*n!- 2* round((n+2)*n!/exp(1)): s := (a,b,n)-> a*aseq(n) + b*bseq( n): seq(s(1,0,n),n = 0..20);  # Gary Detlefs, Dec 11 2018

terms = 24;
CoefficientList[(3E^-z+z-2)/(1-z)^2 + O[z]^terms, z] Range[0, terms-1]! (* Jean-François Alcover, Sep 14 2018 *)

Vec(serlaplace((3*exp(-x + O(x^25)) + x - 2)/(1 - x)^2)) \\ Andrew Howroyd, Jul 10 2018

E.g.f.: (3*exp(-z)+z-2)/(1-z)^2.
a(n) ~ (3*exp(-1) - 1) * n * n!. - Vaclav Kotesovec, Jul 12 2018
a(n) = 3*round((n+2)*n!/e) - (n+2)*n!. - Gary Detlefs, Dec 11 2018
From Seiichi Manyama, Apr 25 2025: (Start)
a(n) = 3 * A000255(n) - n! - (n+1)!.
a(0) = 1, a(1) = 0; a(n) = n*a(n-1) + (n-1)*a(n-2). (End)

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A288950 Number of relaxed compacted binary trees of right height at most one with empty initial and final sequence on level 0.

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A288953 Number of relaxed compacted binary trees of right height at most one with minimal sequences between branch nodes except after the last branch node on level 0.

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A288954 Number of relaxed compacted binary trees of right height at most one with minimal sequences between branch nodes except before the first and after the last branch node on level 0.

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A316666 Number of simple relaxed compacted binary trees of right height at most one with no sequences on level 1 and no final sequences on level 0.

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