A123023 -id:A123023 - cp's OEIS frontend

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.

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)).

A211871 Number T(n,k) of permutations of n elements with no fixed points and largest cycle of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 3, 0, 6, 0, 0, 0, 20, 0, 24, 0, 0, 15, 40, 90, 0, 120, 0, 0, 0, 210, 420, 504, 0, 720, 0, 0, 105, 1120, 2520, 2688, 3360, 0, 5040, 0, 0, 0, 4760, 15120, 27216, 20160, 25920, 0, 40320, 0, 0, 945, 25200, 126000, 193536, 226800, 172800, 226800, 0, 362880

Offset: 0

Alois P. Heinz, Feb 12 2013

From Steven Finch, Sep 27 2021: (Start)
A permutation without fixed points is called a derangement.
For the statistic "length of the smallest cycle", see A348075. (End)

			T(0,0) = 1: (), the empty permutation.
T(2,2) = 1: (2,1).
T(3,3) = 2: (2,3,1), (3,1,2).
T(4,2) = 3: (2,1,4,3), (3,4,1,2), (4,3,2,1).
T(4,4) = 6: (2,4,1,3), (2,3,4,1), (3,1,4,2), (3,4,2,1), (4,1,2,3), (4,3,1,2).
Triangle T(n,k) begins:
  1;
  0,  0;
  0,  0,   1;
  0,  0,   0,    2;
  0,  0,   3,    0,    6;
  0,  0,   0,   20,    0,   24;
  0,  0,  15,   40,   90,    0,  120;
  0,  0,   0,  210,  420,  504,    0, 720;
  0,  0, 105, 1120, 2520, 2688, 3360,   0, 5040;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Steven Finch, Permute, Graph, Map, Derange, arXiv:2111.05720 [math.CO], 2021.
D. Panario and B. Richmond, Exact largest and smallest size of components, Algorithmica, 31 (2001), 413--432.

Columns k=0-3 give: A000007, A000004, A123023 for n>0, A211880.
Row sums give A000166.
Diagonal gives: A000142(n-1) for n>1.
T(n,0) + T(n,2) + T(n,3) gives A055814(n).
Cf. A348075.

A:= proc(n,k) option remember; `if`(n<0, 0, `if`(n=0, 1,
       add(mul(n-i, i=1..j-1)*A(n-j,k), j=2..k)))
    end:
T:= (n, k)-> A(n, k) -`if`(k=0,0,A(n, k-1)):
seq(seq(T(n,k), k=0..n), n=0..12);

A[n_, k_] := A[n, k] = If[n < 0, 0, If[n == 0, 1,
     Sum[Product[n-i, {i, 1, j-1}]*A[n-j, k], {j, 2, k}]]];
T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k-1]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

E.g.f. of column k>1: (exp(x^k/k)-1) * exp(Sum_{j=2..k-1} x^j/j); e.g.f. of column k<=1: 1-k.

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)) ).

A359760 Triangle read by rows. The Kummer triangle, the coefficients of the Kummer polynomials. K(n, k) = binomial(n, k) * oddfactorial(k/2) if k is even, otherwise 0, where oddfactorial(z) := (2z)!/(2^zz!).

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 6, 0, 3, 1, 0, 10, 0, 15, 0, 1, 0, 15, 0, 45, 0, 15, 1, 0, 21, 0, 105, 0, 105, 0, 1, 0, 28, 0, 210, 0, 420, 0, 105, 1, 0, 36, 0, 378, 0, 1260, 0, 945, 0, 1, 0, 45, 0, 630, 0, 3150, 0, 4725, 0, 945, 1, 0, 55, 0, 990, 0, 6930, 0, 17325, 0, 10395, 0

Offset: 0

Peter Luschny, Jan 13 2023

The Kummer numbers K(n, k) are a refinement of the oddfactorial numbers (A001147) in the sense that they are the coefficients of polynomials K(n, x) = Sum_{n..k} K(n, k) * x^k that take the value oddfactorial(n) at x = 1. The coefficients of x^n are the aerated oddfactorial numbers A123023.

These numbers appear in many different versions (see the crossrefs). They are the coefficients of the Chebyshev-Hermite polynomials in signed form when ordered in decreasing powers. Our exposition is based on the seminal paper by Kummer, which preceded the work of Chebyshev and Hermite for more than 20 years. They are also referred to as Bessel numbers of the second kind (Mansour et al.) when the odd powers are omitted.

			Triangle K(n, k) starts:
 [0] 1;
 [1] 1, 0;
 [2] 1, 0,  1;
 [3] 1, 0,  3, 0;
 [4] 1, 0,  6, 0,   3;
 [5] 1, 0, 10, 0,  15, 0;
 [6] 1, 0, 15, 0,  45, 0,   15;
 [7] 1, 0, 21, 0, 105, 0,  105, 0;
 [8] 1, 0, 28, 0, 210, 0,  420, 0, 105;
 [9] 1, 0, 36, 0, 378, 0, 1260, 0, 945, 0;

John Riordan, Introduction to Combinatorial Analysis, Dover (2002), pp. 85-86.

Pierre Humbert, Monographie des polynômes de Kummer, Nouvelles annales de mathématiques, journal des candidats aux écoles polytechnique et normale, Serie 5, Volume 1 (1922), pp. 81-92.
E. E. Kummer, Über die hypergeometrische Reihe, Journal für die reine und angewandte Mathematik 15 (1836): 39-83.
T. Mansour, M. Schork and M. Shattuck, The Generalized Stirling and Bell Numbers Revisited, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3.
Ladislav Truksa, Hypergeometric orthogonal systems of polynomials III, Aktuárské vědy, Vol. 2 (1931), No. 4, 177-203, (see p.200).

Variants: Signed version: A073278. Other variants are the irregular triangle A100861 with zeros deleted, A066325 and A099174 with reversed rows, A111924, A144299, A104556.
Cf. A000085, A001147, A123023, A047974, A115329, A002522, A056107, A359739,
A001879, A001464, A005425, A344501, A123022, A359761.

oddfactorial := proc(z) (2*z)! / (2^z*z!) end:
K := (n, k) -> ifelse(irem(k, 2) = 1, 0, binomial(n, k) * oddfactorial(k/2)):
seq(seq(K(n, k), k = 0..n), n = 0..11);
# Alternative, as coefficients of polynomials:
p := (n, x) -> 2^(n/2)*(-1/x^2)^(-n/2)*KummerU(-n/2, 1/2, -1/(2*x^2)):
seq(print(seq(coeff(simplify(p(n, x)), x, k), k = 0..n)), n = 0 ..9);
# Using the exponential generating function:
egf := exp(x + (t*x)^2 / 2): ser := series(egf, x, 12):
seq(print(seq(coeff(n! * coeff(ser, x, n), t, k), k = 0..n)), n = 0..9);

K[n_, k_] := K[n, k] = Which[OddQ[k], 0, k == 0, 1, n == k, K[n - 1, n - 2], True, K[n - 1, k] n/(n - k)];
Table[K[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 25 2023 *)

from functools import cache
@cache
def K(n: int, k: int) -> int:
    if k %  2: return 0
    if n <  3: return 1
    if n == k: return K(n - 1, n - 2)
    return (K(n - 1, k) * n) // (n - k)
for n in range(10): print([K(n, k) for k in range(n + 1)])

Let p(n, x) = 2^(n/2)*(-1/x^2)^(-n/2)*KummerU(-n/2, 1/2, -1/(2*x^2)).
p(n, 1) = A000085(n); p(n, sqrt(2)) = A047974(n); p(n, 2) = A115329(n);
p(2, n) = A002522(n) (n >= 1); p(3, n) = A056107(n) (n >= 1);
p(n, n) = A359739(n) (n >= 1); 2^n*p(n, 1/2) = A005425(n).
K(n, k) = [x^k] p(n, x).
K(n, k) = [t^k] (n! * [x^n] exp(x + (t*x)^2 / 2)).
K(n, n) = A123023(n).
K(n, n-1) = A123023(n + 1).
K(2*n, 2*n) = A001147(n).
K(4*n, 2*n) = A359761, the central terms without zeros.
K(2*n+2, 2*n) = A001879.
Sum_{k=0..n} (-1)^n * i^k * K(n, k) = A001464(n), ((the number of even involutions) - (the number of odd involutions) in the symmetric group S_n (Robert Israel)).
Sum_{k=0..n} Sum_{j=0..k} K(n, j) = A000085(n + 1).
For a recursion see the Python program.

A380364 Number of rooted combinatorial maps with n edges and without faces of degree 1.

Original entry on oeis.org

1, 1, 4, 30, 284, 3240, 43282, 662760, 11446844, 220193310, 4669558564, 108251161920, 2723857695362, 73941952968000, 2154117314613604, 67038931862069790, 2219781607638887804, 77922680046440538600, 2890682855602209593362, 112998995448368143038120, 4642614436461699746566364

Offset: 0

Andrew Howroyd, Jan 28 2025

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A379433 (planar), A380365 (sensed), A380366 (unsensed).

Cf. A000166, A123023.

seq(n)={my(A=O(x^(2*n+1)), g=serconvol(exp(x^2/2 + A),exp(-x + A)/(1-x))); Vec(substpol(1 + x*deriv(log(serlaplace(g))), x^2, x))}

A211880 Number of permutations of n elements with no fixed points and largest cycle of length 3.

Original entry on oeis.org

0, 0, 0, 2, 0, 20, 40, 210, 1120, 4760, 25200, 157850, 800800, 5345340, 35035000, 222472250, 1648046400, 12000388400, 88529240800, 720929459250, 5786188408000, 48072795270500, 424300329453000, 3731123025279650, 34083741984292000, 323768324084205000

Offset: 0

Alois P. Heinz, Feb 13 2013

nonn

a(n) = A055814(n) - A123023(n). - Vaclav Kotesovec, Oct 09 2013

			a(3) = 2: (2,3,1), (3,1,2).

Alois P. Heinz, Table of n, a(n) for n = 0..250

Column k=3 of A211871.

Cf. A055814, A123023.

egf:= (exp(x^3/3)-1)*exp(x^2/2):
a:= n-> n! *coeff(series(egf, x, n+1), x, n):
seq(a(n), n=0..30);

A[n_, k_] := A[n, k] = If[n < 0, 0, If[n == 0, 1,
     Sum[Product[n - i, {i, 1, j - 1}] A[n - j, k], {j, 2, k}]]];
T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k - 1]];
a[n_] := T[n, 3];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Sep 03 2021, after Alois P. Heinz in A211871 *)

E.g.f.: (exp(x^3/3)-1) * exp(x^2/2).

Recurrence: (n-3)*a(n) = (n-1)*(2*n-5)*a(n-2) + (n-3)*(n-2)*(n-1)*a(n-3) - (n-3)*(n-2)*(n-1)*a(n-4) - (n-4)*(n-3)*(n-2)*(n-1)*a(n-5). - Vaclav Kotesovec, Oct 09 2013

A333158 Irregular triangle read by rows: T(n,k) is the number of k-regular graphs on n labeled nodes with loops allowed, n >= 1, 0 <= k <= n + 1.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 1, 0, 2, 0, 1, 1, 3, 8, 8, 3, 1, 1, 0, 38, 0, 38, 0, 1, 1, 15, 208, 730, 730, 208, 15, 1, 1, 0, 1348, 0, 20670, 0, 1348, 0, 1, 1, 105, 10126, 188790, 781578, 781578, 188790, 10126, 105, 1, 1, 0, 86174, 0, 37885204, 0, 37885204, 0, 86174, 0, 1

Offset: 1

Andrew Howroyd, Mar 09 2020

A loop adds 2 to the degree of its vertex.

			Triangle begins:
  1,   0,     1;
  1,   1,     1,      1;
  1,   0,     2,      0,      1;
  1,   3,     8,      8,      3,      1;
  1,   0,    38,      0,     38,      0,      1;
  1,  15,   208,    730,    730,    208,     15,     1;
  1,   0,  1348,      0,  20670,      0,   1348,     0,   1;
  1, 105, 10126, 188790, 781578, 781578, 188790, 10126, 105, 1;
  ...

Row sums are A322635.
Columns k=0..4 are A000012, A123023, A108246, A110039 (with interspersed zeros), A228697.
Cf. A059441, A333157.

T(n,k) = T(n, n+1-k).

A368213 Triangular array read by rows: Number of permutations of [n] that factor into exactly k-cycles, ordered by n (rows) and divisors k of n (columns).

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 3, 0, 6, 1, 0, 0, 0, 24, 1, 15, 40, 0, 0, 120, 1, 0, 0, 0, 0, 0, 720, 1, 105, 0, 1260, 0, 0, 0, 5040, 1, 0, 2240, 0, 0, 0, 0, 0, 40320, 1, 945, 0, 0, 72576, 0, 0, 0, 0, 362880, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3628800, 1, 10395, 246400, 1247400, 0, 6652800, 0, 0, 0, 0, 0, 39916800

Offset: 1

Marko Riedel, Dec 17 2023

			Row n=6 is 1, 15, 40, 120 because there is one permutation of [6] consisting of six fixed points, there are 15 permutations consisting of three transpositions, there are forty permutations consisting of two three-cycles and there are one hundred and twenty permutations consisting of just one six-cycle (6!/6).
Triangular array starts:
[ 1] 1;
[ 2] 1,   1;
[ 3] 1,   0,    2;
[ 4] 1,   3,    0,    6;
[ 5] 1,   0,    0,    0,    24;
[ 6] 1,  15,   40,    0,     0, 120;
[ 7] 1,   0,    0,    0,     0,   0, 720;
[ 8] 1, 105,    0, 1260,     0,   0,   0, 5040;
[ 9] 1,   0, 2240,    0,     0,   0,   0,    0, 40320;
[10] 1, 945,    0,    0, 72576,   0,   0,    0,     0, 362880;

P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University Press, 2009, pages 120-122.

Marko Riedel, math.stackexchange.com, Number of permutations in the symmetric group.
Marko Riedel, Computing the number of permutations of [n] with k m-cycles with analytic combinatorics.
Marko Riedel, Computing the number of permutations of [n] with k m-cycles using the principle of inclusion-exclusion.

Cf. A005225 (row sums), A008290.

Cf. A123023 (column 2), A052502 (column 3), A060706 (column 4).

T:= (n, m)-> `if`(irem(n,m)=0, n!/m^(n/m)/(n/m)!, 0):
seq(seq(T(n, m), m = 1..n), n=1..15);

A368213[n_,k_]:=If[Divisible[n,k],n!/(k^(n/k)(n/k)!),0];
Table[A368213[n,k],{n,15},{k,n}] (* Paolo Xausa, Dec 18 2023 *)

def T(n, d): return factorial(n) // (d ** (n//d) * factorial(n//d))
for n in range(1, 19):
    print([T(n, d) if n % d == 0 else 0 for d in range(1, n+1)])
# Peter Luschny, Dec 17 2023

T(n, k) = n! / ( k^(n/k) * (n/k)! ) if k divides n otherwise 0.

A227937 Partitions of n labeled elements into subsets of two or three elements.

Original entry on oeis.org

1, 0, 1, 1, 3, 10, 25, 105, 385, 1540, 7245, 32725, 164395, 870870, 4689685, 27152125, 161786625, 997196200, 6443061625, 42702885225, 292938721075, 2078239413250, 15119319039825, 113390111659825, 873538909468225, 6894294734827500, 55855506234653125, 463151808682688125, 3927996120260086875, 34081631999814148750, 301951521812713898125, 2731127272307562253125, 25208456056107710010625, 237164027532948085570000

Offset: 0

David Eppstein, Oct 06 2013

Periodic modulo two and modulo three, but appears to eventually be divisible by other prime powers.

			The five elements a, b, c, d, e have ten partitions into sets of size two or three: ab/cde, ac/bde, ad/bce, ae/bcd, bc/ade, bd/ace, be/acd, cd/abe, ce/abd, and de/abc.

Cf. A000296, A123023.

Flatten[{1,RecurrenceTable[{2*a[n] - 2*(n-1)*a[n-2]-(n-2)*(n-1)*a[n-3] == 0,a[1]==0,a[2]==1,a[3]==1}, a, {n, 1, 20}]}] (* Vaclav Kotesovec, Oct 09 2013 *)

x='x+O('x^66); Vec( serlaplace( exp( x^2/2 + x^3/6 ) ) ) \\ Joerg Arndt, Oct 07 2013

a(n) = (n-1)*a(n-2) + (n-1)*(n-2)*a(n-3)/2.
E.g.f.:  exp( x^2/2 + x^3/6 ). [Joerg Arndt, Oct 07 2013]
a(n) ~ n^(2*n/3) * 2^(-n/3) * exp(2/9 - 2*n/3 - (2*n)^(1/3)/3 + (2*n)^(2/3)/2)/sqrt(3) * (1 + 34/(162*(2*n)^(1/3)) - 34802/(131220*(2*n)^(2/3))). - Vaclav Kotesovec, Oct 09 2013

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)

cp's OEIS Frontend

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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A211871 Number T(n,k) of permutations of n elements with no fixed points and largest cycle of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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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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A359760 Triangle read by rows. The Kummer triangle, the coefficients of the Kummer polynomials. K(n, k) = binomial(n, k) * oddfactorial(k/2) if k is even, otherwise 0, where oddfactorial(z) := (2*z)!/(2^z*z!).

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A380364 Number of rooted combinatorial maps with n edges and without faces of degree 1.

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A211880 Number of permutations of n elements with no fixed points and largest cycle of length 3.

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A333158 Irregular triangle read by rows: T(n,k) is the number of k-regular graphs on n labeled nodes with loops allowed, n >= 1, 0 <= k <= n + 1.

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A368213 Triangular array read by rows: Number of permutations of [n] that factor into exactly k-cycles, ordered by n (rows) and divisors k of n (columns).

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A359760 Triangle read by rows. The Kummer triangle, the coefficients of the Kummer polynomials. K(n, k) = binomial(n, k) * oddfactorial(k/2) if k is even, otherwise 0, where oddfactorial(z) := (2z)!/(2^zz!).