A000309 -id:A000309 - cp's OEIS frontend

A000699 Number of irreducible chord diagrams with 2n nodes.

1, 1, 1, 4, 27, 248, 2830, 38232, 593859, 10401712, 202601898, 4342263000, 101551822350, 2573779506192, 70282204726396, 2057490936366320, 64291032462761955, 2136017303903513184, 75197869250518812754, 2796475872605709079512, 109549714522464120960474, 4509302910783496963256400, 194584224274515194731540740

Offset: 0

N. J. A. Sloane

Perturbation expansion in quantum field theory: spinor case in 4 spacetime dimensions.
a(n)*2^(-n) is the coefficient of the x^(2*n-1) term in the series reversal of the asymptotic expansion of 2*DawsonF(x) = sqrt(Pi)*exp(-x^2)*erfi(x) for x -> inf. - Vladimir Reshetnikov, Apr 23 2016
The September 2018 talk by Noam Zeilberger (see link to video) connects three topics (planar maps, Tamari lattices, lambda calculus) and eight sequences: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827. - N. J. A. Sloane, Sep 17 2018
A set partition is topologically connected if the graph whose vertices are the blocks and whose edges are crossing pairs of blocks is connected, where two blocks cross each other if they are of the form {{...x...y...},{...z...t...}} for some x < z < y < t or z < x < t < y. Then a(n) is the number of topologically connected 2-uniform set partitions of {1...2n}. See my links for examples. - Gus Wiseman, Feb 23 2019
From Julien Courtiel, Oct 09 2024: (Start)
a(n) is the number of rooted bridgeless combinatorial maps with n edges (genus is not fixed). A map is bridgeless if it has no edge whose removal disconnects the graph. For example, for n=2, there are 4 bridgeless maps with 2 edges: 2 planar maps with 1 vertex (either two consecutive loops, or two nested loops), 1 toric map with 1 vertex, and 1 planar map with 2 vertices connected by a double edge.
Also, a(n) is the number of trees with n edges equipped with a binary tubing. A tube is a connected subgraph. A binary tubing of a tree is a nested set collection S of tubes such that 1. S contains the tube of all vertices 2. Every tube of S is either reduced to one vertex, or it can be can partitioned by 2 tubes of S.
(End)

			a(31)=627625976637472254550352492162870816129760 was computed using Kreimer's Hopf algebra of rooted trees. It subsumes 2.6*10^21 terms in quantum field theory.
G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 27*x^4 + 248*x^5 + 2830*x^6 +...
where d/dx (A(x) - 1)^2/x = 1 + 4*x + 27*x^2 + 248*x^3 + 2830*x^4 +...

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..404 (terms up to n=100 from T. D. Noe)
Serban T. Belinschi, Marek Bozejko, Franz Lehner, and Roland Speicher, The normal distribution is "boxplus"-infinitely divisible, arXiv:0910.4263 [math.OA], 2009-2010.
Daniel J. Bernstein, S. Engels, T. Lange, R. Niederhagen, et al., Faster elliptic-curve discrete logarithms on FPGAs, Preprint, 2016.
Michael Borinsky, Generating asymptotics for factorially divergent sequences, arXiv preprint arXiv:1603.01236 [math.CO], 2016.
Michael Borinsky, Generating asymptotics for factorially divergent sequences, El. J. Combin. 25 (2018) P4.1, Sect 7.1.
Michael Borinsky, Gerald V. Dunne, and Karen Yeats, Tree-tubings and the combinatorics of resurgent Dyson-Schwinger equations, arXiv:2408.15883 [math-ph], 2024. See p. 9.
David J. Broadhurst and Dirk Kreimer, Combinatoric explosion of renormalization ..., arXiv:hep-th/9912093, 1999, 2000.
David J. Broadhurst and Dirk Kreimer, Combinatoric explosion of renormalization tamed by Hopf algebra: 30-loop Pad-Borel resummation, Phys. Lett. B 475 (2000), 63-70.
Christian Brouder, On the trees of quantum fields, arXiv:hep-th/9906111, 1999, p. 6.
Jonathan Burns, Assembly Graph Words - Single Transverse Component (Counts).
Jonathan Burns, Egor Dolzhenko, Natasa Jonoska, Tilahun Muche and Masahico Saito, Four-Regular Graphs with Rigid Vertices Associated to DNA Recombination, May 23, 2011.
Jonathan Burns and Tilahun Muche, Counting Irreducible Double Occurrence Words, arXiv preprint arXiv:1105.2926 [math.CO], 2011.
Julien Courtiel, Karen Yeats, and Noam Zeilberger, Connected chord diagrams and bridgeless maps, arXiv:1611.04611 [math.CO], 2016.
Serge Dulucq, Etude combinatoire de problèmes d'énumération, d'algorithmique sur les arbres et de codage par des mots, a thesis presented to l'Université de Bordeaux I, 1987. (Annotated scanned copy)
Philippe Flajolet and Marc Noy, Analytic Combinatorics of Chord Diagrams, in: Formal power series and algebraic combinatorics (FPSAC '00) Moscow, 2000, pp. 191-201.
Zhen Huang, Denis Golež, Hugo U. R. Strand, and Jason Kaye, Automated evaluation of imaginary time strong coupling diagrams by sum-of-exponentials hybridization fitting, arXiv:2503.19727 [cond-mat.str-el], 2025. See p. 21.
Martin Klazar, Non-P-recursiveness of numbers of matchings or linear chord diagrams with many crossings, Advances in Appl. Math., Vol. 30 (2003), pp. 126-136.
Martin Klazar, Counting even and odd partitions, Amer. Math. Monthly, 110 (No. 6, 2003), 527-532.
Florian Kogelbauer and Ilya Karlin, On the Relation of Exact Hydrodynamics to the Chapman-Enskog Series, arXiv:2506.17441 [math-ph], 2025. See pp. 4-5.
Ali Assem Mahmoud, On the Asymptotics of Connected Chord Diagrams, University of Waterloo (Ontario, Canada 2019).
Ali Assem Mahmoud, An Asymptotic Expansion for the Number of 2-Connected Chord Diagrams, arXiv:2009.12688 [math.CO], 2020.
Ali Assem Mahmoud, Chord Diagrams and the Asymptotic Analysis of QED-type Theories, arXiv:2011.04291 [hep-th], 2020.
Ali Assem Mahmoud, An asymptotic expansion for the number of two-connected chord diagrams, J. Math. Phys. (2023) Vol. 64, 122301. See Example 2.1.
Ali Assem Mahmoud and Karen Yeats, Connected Chord Diagrams and the Combinatorics of Asymptotic Expansions, arXiv:2010.06550 [math.CO], 2020.
Nicolas Marie and Karen Yeats, A chord diagram expansion coming from some Dyson-Schwinger equations, arXiv:1210.5457 [math.CO], 2012, Section 4.1.
Albert Nijenhuis and Herbert S. Wilf, The enumeration of connected graphs and linked diagrams, J. Combin. Theory Ser. A 27 (1979), no. 3, 356--359. MR0555804 (82b:05074).
Vincent Pilaud and Juanjo Rué, Analytic combinatorics of chord and hyperchord diagrams with k crossings, arXiv preprint arXiv:1307.6440 [math.CO], 2013.
J. Riordan, Letter, Jul 06 1978
Einar A. Rødland, Pseudoknots in RNA Secondary Structures: Representation, Enumeration, and Prevalence, J. Comput. Biology, Vol 13, No 6 (2006), 1197-1213. (see equation 10)
R. R. Stein, On a class of linked diagrams, I. Enumeration, J. Combin. Theory, A 24 (1978), 357-366.
R. R. Stein and C. J. Everett, On a class of linked diagrams, II. Asymptotics, Discrete Math., 21 (1978), 309-318.
Jacques Touchard, Sur un problème de configurations et sur les fractions continues, Canad. J. Math., 4 (1952), 2-25.
Jacques Touchard, Sur un problème de configurations et sur les fractions continues, Canad. J. Math., 4 (1952), 2-25. [Annotated, corrected, scanned copy]
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus. III: Nonseparable maps, J. Combinatorial Theory Ser. B 18 (1975), 222-259 (nonseparable integer systems on n pairs). (Give an incorrect a(6)=2720.)
Gus Wiseman, The a(4) = 27 connected chord diagrams.
Gus Wiseman, The a(5) = 248 connected chord diagrams.
Gus Wiseman, Constructive Mathematica program for A000699.
Noam Zeilberger, A theory of linear typings as flows on 3-valent graphs, arXiv:1804.10540 [cs.LO], 2018.
Noam Zeilberger, A Sequent Calculus for a Semi-Associative Law, arXiv:1803.10080 [math.LO], 2018-2019 (A revised version of a 2017 conference paper).
Noam Zeilberger, A proof-theoretic analysis of the rotation lattice of binary trees, Part 1 (video), Rutgers Experimental Math Seminar, Sep 13 2018. Part 2 is vimeo.com/289910554.

Sequences mentioned in the Noam Zeilberger 2018 video: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827.
Cf. A004300, A051862, A212273. Column sums of A232223. First column of A322402.
Cf. A007297, A016098, A099947, A136653, A268815, A306438, A324166, A324172, A324173, A324327.

A000699 := proc(n)
    option remember;
    if n <= 1 then
        1;
    else
        add((2*i-1)*procname(i)*procname(n-i),i=1..n-1) ;
    end if;
end proc:
seq(A000699(n),n=0..30) ; # R. J. Mathar, Jun 12 2018

terms = 22; A[] = 0; Do[A[x] = x + x^2 * D[A[x]^2/x, x] + O[x]^(terms+1) // Normal, terms]; CoefficientList[A[x], x] // Rest (* Jean-François Alcover, Apr 06 2012, after Paul D. Hanna, updated Jan 11 2018 *)
a = ConstantArray[0,20]; a[[1]]=1; Do[a[[n]] = (n-1)*Sum[a[[i]]*a[[n-i]],{i,1,n-1}],{n,2,20}]; a (* Vaclav Kotesovec, Feb 22 2014 *)
Module[{max = 20, s}, s = InverseSeries[ComplexExpand[Re[Series[2 DawsonF[x], {x, Infinity, 2 max + 1}]]]]; Table[SeriesCoefficient[s, 2 n - 1] 2^n, {n, 1, max}]] (* Vladimir Reshetnikov, Apr 23 2016 *)

{a(n)=local(A=1+x*O(x^n)); for(i=1, n, A=1+x+x^2*deriv((A-1)^2/x)+x*O(x^n)); polcoeff(A, n)} \\ Paul D. Hanna, Dec 31 2010 [Modified to include a(0) = 1. - Paul D. Hanna, Nov 06 2020]

{a(n) = my(A); A = 1+O(x) ; for( i=0, n, A = 1+x + (A-1)*(2*x*A' - A + 1)); polcoeff(A, n)}; /* Michael Somos, May 12 2012 [Modified to include a(0) = 1. - Paul D. Hanna, Nov 06 2020] */

seq(N) = {
  my(a = vector(N)); a[1] = 1;
  for (n=2, N, a[n] = sum(k=1, n-1, (2*k-1)*a[k]*a[n-k])); a;
};
seq(22)  \\ Gheorghe Coserea, Jan 22 2017

seq(n)={my(g=serlaplace(1 / sqrt(1 - 2*x + O(x*x^n)))); Vec(sqrt((x/serreverse( x*g^2 ))))} \\ Andrew Howroyd, Nov 21 2024

def A000699_list(n):
    list = [1, 1] + [0] * (n - 1)
    for i in range(2, n + 1):
        list[i] = (i - 1) * sum(list[j] * list[i - j] for j in range(1, i))
    return list
print(A000699_list(22)) # M. Eren Kesim, Jun 23 2021

a(n) = (n-1)*Sum_{i=1..n-1} a(i)*a(n-i) for n > 1, with a(1) = a(0) = 1. [Modified to include a(0) = 1. - Paul D. Hanna, Nov 06 2020]
A212273(n) = n * a(n). - Michael Somos, May 12 2012
G.f. satisfies: A(x) = 1 + x + x^2*[d/dx (A(x) - 1)^2/x]. - Paul D. Hanna, Dec 31 2010 [Modified to include a(0) = 1. - Paul D. Hanna, Nov 06 2020]
a(n) ~ n^n * 2^(n+1/2) / exp(n+1) * (1 - 31/(24*n) - 2207/(1152*n^2) - 3085547/(414720*n^3) - 1842851707/(39813120*n^4) - ...). - Vaclav Kotesovec, Feb 22 2014, extended Oct 23 2017
G.f. A(x) satisfies: 1 = A(x) - x/(A(x) - 2*x/(A(x) - 3*x/(A(x) - 4*x/(A(x) - 5*x/(A(x) - ...))))), a continued fraction relation. - Paul D. Hanna, Nov 04 2020
G.f. A(x) satisfies: A(x*B(x)^2) = B(x) where B(x) is the g.f. of A001147. - Andrew Howroyd, Nov 21 2024

More terms from David Broadhurst, Dec 14 1999
Inserted "chord" in definition. - N. J. A. Sloane, Jan 19 2017
Added a(0)=1. - N. J. A. Sloane, Nov 05 2020
Modified formulas slightly to include a(0) = 1. - Paul D. Hanna, Nov 06 2020

A000698 A problem of configurations: a(0) = 1; for n>0, a(n) = (2n-1)!! - Sum_{k=1..n-1} (2k-1)!! a(n-k). Also the number of shellings of an n-cube, divided by 2^n n!.

Original entry on oeis.org

1, 1, 2, 10, 74, 706, 8162, 110410, 1708394, 29752066, 576037442, 12277827850, 285764591114, 7213364729026, 196316804255522, 5731249477826890, 178676789473121834, 5925085744543837186, 208256802758892355202, 7734158085942678174730

Offset: 0

N. J. A. Sloane, Richard Ehrenborg, Gabor Hetyei

Also number of nonisomorphic unlabeled connected Feynman diagrams of order 2n-2 for the electron propagator of quantum electrodynamics (QED), including vanishing diagrams. [Corrected by Charles R Greathouse IV, Jan 24 2014][Clarified by Robert Coquereaux, Sep 14 2014]
a(n+1) is the moment of order 2*n for the probability density function rho(x) = (1/sqrt(2*Pi))*exp(x^2/2)/[(u(x))^2+Pi/2], with u(x) = Integral_{t=0..x} exp(t*t/2) dt, on the real interval -infinity..infinity. - Groux Roland, Jan 13 2009
Starting (1, 2, 10, 74, ...) = INVERTi transform of A001147: (1, 3, 15, 105, ...). - Gary W. Adamson, Oct 21 2009
The Cvitanovic et al. paper relates this sequence to A005411 and A005413. - Robert Munafo, Jan 24 2010
Hankel transform of a(n+1) is A168467. - Paul Barry, Nov 26 2009
a(n) = number of labeled Dyck (n-1)-paths (A000108) in which each vertex that terminates an upstep is labeled with an integer i in [0,h], where h is the height of the vertex . For example UDUD contributes 4 labeled paths--0D0D, 0D1D, 1D0D, 1D1D where upsteps are replaced by their labels--and UUDD contributes 6 labeled paths to a(3)=10. The Deléham (Mar 24 2007) formula below counts these labeled paths by number of "0" labels. - David Callan, Aug 23 2011
a(n) is the number of indecomposable perfect matchings on [2n]. A perfect matching on [2n] is decomposable if a nonempty subset of the edges forms a perfect matching on [2k] for some kDavid Callan, Nov 29 2012
From Robert Coquereaux, Sep 12 2014: (Start)
QED diagrams are graphs with two kinds of edges (lines): a (non-oriented), f (oriented), and only one kind of (internal) vertex: aff. They may have internal and external (i.e., pendant) lines. The order is the number of (internal) vertices. Vanishing diagrams: QED diagrams containing loops of type f with an odd number of vertices are set to 0 (Furry theorem). Proper diagrams: connected QED diagrams that remain connected when an arbitrary internal line is cut.
The number of Feynman diagrams of order 2n for the electron propagator (2-point function of QED), vanishing or not, proper or not, of order 2n, starting from n = 0, is given by 1, 2, 10, 74, 706, 8162, ..., i.e., this sequence A000698, with the first term (equal to 1) dropped. Call Sf the associated g.f.
The number of non-vanishing Feynman diagrams, for the same 2-point function, is given by 1, 1, 4, 25, 208, 2146, ..., i.e., by the sequence A005411, with a first term of order 0, equal to 1, added. Call S the associated g.f.
If one does not remove the vanishing diagram, but, at the same time, considers only those graphs that are proper, one obtains the Feynman diagrams (vanishing and non-vanishing) for the self-energy function of QED, 0, 1, 3, 21, 207, 2529, ..., i.e., the sequence A115974 with a first term of order 0, equal to 0, added. A115974 is twice A167872. Call Sigmaf the associated g.f.
If one removes the vanishing diagrams and, at the same time, considers only those graphs that are proper, one obtains the Feynman diagrams for the self-energy function of QED given by 0, 1, 3, 18, 153, 1638, ..., i.e., by the sequence A005412, with a first term of order 0, equal to 0, added. Call Sigma the associated g.f.
Then Sf = 1/(1-Sigmaf) and S = 1/(1-Sigma). (End)
For n>0 sum over all Dyck paths of semilength n-1 of products over all peaks p of (x_p+y_p)/y_p, where x_p and y_p are the coordinates of peak p. - Alois P. Heinz, May 22 2015
Also, counts certain isomorphism classes of closed normal linear lambda terms. [N. Zeilberger, 2015]. - N. J. A. Sloane, Sep 18 2016
The September 2018 talk by Noam Zeilberger (see link to video) connects three topics (planar maps, Tamari lattices, lambda calculus) and eight sequences: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827. - N. J. A. Sloane, Sep 17 2018
For n >= 2, a(n) is the number of coalescent histories for a pair consisting of a matching lodgepole gene tree and species tree with 2n-1 leaves. - Noah A Rosenberg, Jun 21 2022

			G.f. = 1 + x + 2*x^2 + 10*x^3 + 74*x^4 + 706*x^5 + 8162*x^6 + 110410*x^7 + ...

Dubois C., Giorgetti A., Genestier R. (2016) Tests and Proofs for Enumerative Combinatorics. In: Aichernig B., Furia C. (eds) Tests and Proofs. TAP 2016. Lecture Notes in Computer Science, vol 9762. Springer.
R. W. Robinson, Counting irreducible Feynman diagrams exactly and asymptotically, Abstracts Amer. Math. Soc., 2002, #975-05-270.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
D. Arques and J.-F. Beraud, Rooted maps on orientable surfaces, Riccati's equation and continued fractions, Discrete Math., 215 (2000), 1-12.
F. Battaglia and T. F. George, A Pascal type triangle for the number of topologically distinct many-electron Feynman diagrams, J. Math. Chem. 2 (1988) 241-247.
S. Birdsong and G. Hetyei, A Gray Code for the Shelling Types of the Boundary of a Hypercube, arXiv preprint arXiv:1111.4710 [math.CO], 2011.
S. Birdsong and G. Hetyei, A Gray Code for the Shelling Types of the Boundary of a Hypercube, Discrete Math. 313 (2013), no. 3, 258-268. MR2995390.
Jonathan Burns, Assembly Graph Words - Single Transverse Component (Counts)
Jonathan Burns and Tilahun Muche, Counting Irreducible Double Occurrence Words, arXiv preprint arXiv:1105.2926 [math.CO], 2011, Lemma 3.2.
Jonathan Burns, Egor Dolzhenko, Natasa Jonoska, Tilahun Muche and Masahico Saito, Four-Regular Graphs with Rigid Vertices Associated to DNA Recombination, Disc. Appl. Math. 161 (2013) 1378-1394
J. Courtiel, K. Yeats, and N. Zeilberger, Connected chord diagrams and bridgeless maps, arXiv:1661.04611, Proposition 13.
P. Cvitanovic, B. Lautrup and R. B. Pearson, The number and weights of Feynman diagrams, Phys. Rev. D18, (1978), 1939-1949, (eq 3.34 and fig 2b). DOI:10.1103/PhysRevD.18.1939
M. A. Deryagina and A. D. Mednykh, On the enumeration of circular maps with given number of edges, Siberian Mathematical Journal, 54, No. 6, 2013, 624-639.
F. Disanto and N. A. Rosenberg, Coalescent histories for lodgepole species trees, J. Comput. Biol. 22 (2015), 918-929.
S. Dulucq, Etude combinatoire de problèmes d'énumération, d'algorithmique sur les arbres et de codage par des mots, a thesis presented to l'Université de Bordeaux I, 1987. (Annotated scanned copy)
Trinh Khanh Duy and Tomoyuki Shirai, The mean spectral measures of random Jacobi matrices related to Gaussian beta ensembles, arXiv:1504.06904 [math.SP], 2015.
G. Edgar, Transseries for beginners, arXiv:0801.4877v5 [math.RA], 2008-2009.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
Ali Assem Mahmoud, On the Asymptotics of Connected Chord Diagrams, University of Waterloo (Ontario, Canada 2019).
Ali Assem Mahmoud and Karen Yeats, Connected Chord Diagrams and the Combinatorics of Asymptotic Expansions, arXiv:2010.06550 [math.CO], 2020.
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, arXiv:1103.4936 [math.CO], 2011.
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, Aequat. Math., 80 (2010), 291-318. see p. 292.
R. J. Mathar, Table of Feynman Diagrams of the Interacting Fermion Green's Function, Int. J. Quant. Chem. 107 (2007) 1975. Also on arXiv, arXiv:physics/0512022 [physics.atom-ph], 2015-2016.
R. J. Mathar, Feynman diagrams of the QED vacuum polarization, vixra:1901.0148 (2019), Section VII.
L. G. Molinari, Hedin's equations and enumeration of Feynman diagrams, Phys. Rev. B, 71 (2005), 113102.
A. Prunotto, W. M. Alberico, and P. Czerski, Feynman diagrams and rooted maps, Open Phys. 16 (2018) 149-167.
J. Touchard, Sur un problème de configurations et sur les fractions continues, Canad. J. Math., 4 (1952), 2-25.
J. Touchard, Sur un problème de configurations et sur les fractions continues, Canad. J. Math., 4 (1952), 2-25. [Annotated, corrected, scanned copy]
Wikipedia, Feynman diagram
Noam Zeilberger, Counting isomorphism classes of beta-normal linear lambda terms, arXiv:1509.07596 [cs.LO], 2015.
Noam Zeilberger, A theory of linear typings as flows on 3-valent graphs, arXiv:1804.10540 [cs.LO], 2018.
Noam Zeilberger, A Sequent Calculus for a Semi-Associative Law, arXiv preprint 1803.10030, March 2018 (A revised version of a 2017 conference paper)
Noam Zeilberger, A proof-theoretic analysis of the rotation lattice of binary trees, Part 1 (video), Rutgers Experimental Math Seminar, Sep 13 2018. Part 2 is vimeo.com/289910554.
P. Zinn-Justin and J.-B. Zuber, Matrix integrals and the generation and counting of virtual tangles and links, arXiv:math-ph/0303049, 2003.

Sequences mentioned in the Noam Zeilberger 2018 video: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827.
Cf. A004208, A000165, A001147, A005412, row sums of A053979, A005411, A115974, A167872.
Column k=1 of A258219, A258222.
Row sums of A322398.

A006882 := proc(n) option remember; if n <= 1 then 1 else n*procname(n-2); fi; end;
A000698:=proc(n) option remember; global df; local k; if n=0 then RETURN(1); fi; A006882(2*n-1) - add(A006882(2*k-1)*A000698(n-k),k=1..n-1); end;
A000698 := proc(n::integer) local resul,fac,pows,c,c1,p,i ; if n = 0 then RETURN(1) ; else pows := combinat[partition](n) ; resul := 0 ; for p from 1 to nops(pows) do c := combinat[permute](op(p,pows)) ; c1 := op(1,c) ; fac := nops(c) ; for i from 1 to nops(c1) do fac := fac*doublefactorial(2*op(i,c1)-1) ; od ; resul := resul-(-1)^nops(c1)*fac ; od : fi ; RETURN(resul) ; end; # R. J. Mathar, Apr 24 2006
# alternative Maple program:
b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, b(x-1, y-1, false)*`if`(t, (x+y)/y, 1) +
                   b(x-1, y+1, true)  ))
    end:
a:= n-> `if`(n=0, 1, b(2*n-2, 0, false)):
seq(a(n), n=0..25);  # Alois P. Heinz, May 23 2015
a_list := proc(len) local n, A; if len=1 then return [1] fi: A := Array(-1..len-2); A[-1] := 1; A[0] := 1; for n to len-2 do A[n] := (2*n-1)*A[n-1]+add(A[j]*A[n-j-1], j=0..n-1) od: convert(A, list) end: a_list(20); # Peter Luschny, Jul 18 2017

a[n_] := a[n] = (2n - 1)!! - Sum[ a[n - k](2k - 1)!!, {k, n-1}]; Array[a, 18, 0] (* Ignacio D. Peixoto, Jun 23 2006 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ 2 - 1 / Sum[ (2 k - 1)!! x^k, {k, 0, n}], {x, 0, n}]]; (* Michael Somos, Nov 16 2011 *)
a[n_]:= SeriesCoefficient[1+x(1/x+(E^((1/2)/x) Sqrt[2/\[Pi]] Sqrt[-(1/x)])/Erfc[Sqrt[-(1/x)]/Sqrt[2]]), {x,0,n}, Assumptions -> x >0](* Robert Coquereaux, Sep 14 2014 *)
max = 20; g = t/Fold[1 - ((t + #2)*z)/#1 &, 1, Range[max, 1, -1]]; T[n_, k_] := SeriesCoefficient[g, {z, 0, n}, {t, 0, k}]; a[0] = 1; a[n_] := Sum[T[n-1, k], {k, 0, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jan 31 2016, after Philippe Deléham *)

{a(n) = if( n<0, 0, polcoeff( 2 - 1 / sum( k=0, n, x^k * (2*k)! /(2^k * k!), x * O(x^n)), n))}; /* Michael Somos, Feb 08 2011 */

{a(n) = my(A); if( n<1, n==0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (2*k - 3) * A[k-1] + sum( j=1, k-1, A[j] * A[k-j])); A[n])}; /* Michael Somos, Jul 24 2011 */

from sympy import factorial2, cacheit
@cacheit
def a(n): return 1 if n == 0 else factorial2(2*n - 1) - sum(factorial2(2*k - 1)*a(n - k) for k in range(1, n))
[a(n) for n in range(51)]  # Indranil Ghosh, Jul 18 2017

G.f.: 2 - 1/(1 + Sum_{n>=1} (2*n-1)!! * x^n ).
a(n+1) = Sum_{k=0..n} A089949(n, k)*2^k. - Philippe Deléham, Aug 15 2005
a(n+1) = Sum_{k=0..n} A053979(n,k). - Philippe Deléham, Mar 24 2007
From Paul Barry, Nov 26 2009: (Start)
G.f.: 1+x/(1-2x/(1-3x/(1-4x/(1-5x/(1-6x/(1-... (continued fraction).
G.f.: 1+x/(1-2x-6x^2/(1-7x-20x^2/(1-11x-42x^2/(1-15x-72x^2/(1-19x-110x^2/(1-... (continued fraction). (End)
G.f.: 1 + x * B(x) * C(x) where B(x) is the g.f. for A001147 and C(x) is the g.f. for A005416. - Michael Somos, Feb 08 2011
G.f.: 1+x/W(0); where W(k)=1+x+x*2k-x*(2k+3)/W(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 17 2011
From Peter Bala, Dec 22 2011: (Start)
Recurrence relation: a(n+1) = (2*n-1)*a(n) + Sum_{k = 1..n} a(k)*a(n+1-k) for n >= 0 and a(1) = 1.
The o.g.f. B(x) = Sum_{n>=1} a(n)*x^(2*n-1) = x + 2*x^3 + 10*x^5 + 74*x^7 + ... satisfies the Riccati differential equation y'(x) = -1/x^2 + (1/x^3)*y(x) - (1/x^2)*y(x)^2 with initial condition y(0) = 0 (cf. A005412). The solution is B(x) = 1/z(x) + 1/x, where z(x) = -Sum_{n>=0} A001147(n) * x^(2*n+1) = -(x + x^3 + 3*x^5 + 15*x^7 + ...). The function b(x) = -B(1/x) satisfies b'(x) = -1 - (x + b(x))*b(x). Hence the differential operator (D^2 + x*D + 1), where D = d/dx, factorizes as (D - a(x))*(D - b(x)), where a(x) = -(x + b(x)), as conjectured by [Edgar, Problem 4.32]. For a refinement of this sequence see A053979. (End)
From Sergei N. Gladkovskii, Aug 19 2012, Oct 24 2012, Mar 19 2013, May 20 2013, May 29 2013, Aug 04 2013, Aug 05 2013: (Start)
Continued fractions:
G.f.: 2 - G(0) where G(k) = 1 - (k+1)*x/G(k+1).
G.f.: 2 - U(0) where U(k) = 1 - (2*k+1)*x/(1 - (2*k+2)*x/U(k+1)).
G.f.: 2 - U(0) where U(k) = 1 - (4*k+1)*x - (2*k+1)*(2*k+2)*x^2/U(k+1).
G.f.: 1/Q(0) where Q(k) = 1 - x*(2*k+2)/(1 - x*(2*k+3)/Q(k+1)).
G.f.: 1 + x/Q(0) where Q(k) = 1 - x*(k+2)/Q(k+1).
G.f.: 2 - G(0)/2 where G(k) = 1 + 1/(1 - 2*x*(2*k+1)/(2*x*(2*k+1) - 1 + 2*x*(2*k+2)/ G(k+1))).
G.f.: 1 + x*G(0) where G(k) = 1 - x*(k+2)/(x*(k+2) - 1/G(k+1)).
G.f.: 2 - 1/B(x) where B(x) is the g.f. of A001147.
G.f.: 1 + x/(1-2*x*B(x)) where B(x) is the g.f. of A167872. (End)
a(n) ~ 2^(n+1/2) * n^n / exp(n). - Vaclav Kotesovec, Mar 10 2014
G.f.: 1 + x*(1/x + (sqrt(2/Pi) * exp(1/(2*x)) * sqrt(-1/x))/Erfc(sqrt(-1/x)/sqrt(2))) where Erfc(z) = 1 - Erf(z) is the complementary error function, and Erf(z) is the integral of the Gaussian distribution. This generating function is obtained from the generating functional of (4-dimensional) QED, evaluated in dimension 0 for the 2-point function, without the modification implementing Furry theorem. - Robert Coquereaux, Sep 14 2014
From Peter Bala, May 23 2017: (Start)
G.f. A(x) = 1 + x/(1 + x - 3*x/(1 + 3*x - 5*x/(1 + 5*x - 7*x/(1 + 7*x - ...)))).
A(x) = 1 + x/(1 + x - 3*x/(1 - 2*x/(1 - 5*x/(1 - 4*x/(1 - 7*x/(1 - 6*x/(1 - ...))))))). (End)

Formula corrected by Ignacio D. Peixoto, Jun 23 2006

More terms from Sean A. Irvine, Feb 27 2011

A000260 Number of rooted simplicial 3-polytopes with n+3 nodes; or rooted 3-connected triangulations with 2n+2 faces; or rooted 3-connected trivalent maps with 2n+2 vertices.

Original entry on oeis.org

1, 1, 3, 13, 68, 399, 2530, 16965, 118668, 857956, 6369883, 48336171, 373537388, 2931682810, 23317105140, 187606350645, 1524813969276, 12504654858828, 103367824774012, 860593023907540, 7211115497448720, 60776550501588855

Offset: 0

N. J. A. Sloane

Number of rooted loopless planar maps with n edges. E.g., there are a(2)=3 loopless planar maps with 2 edges: two rooted paths (.-.-.) and one digon (.=.). - Valery A. Liskovets, Sep 25 2003
Number of intervals (i.e., ordered pairs (x,y) such that x<=y) in the Tamari lattice (rotation lattice of binary trees) of size n (see Pallo and Chapoton references). - Ralf Stephan, May 08 2007, Jean Pallo (Jean.Pallo(AT)u-bourgogne.fr), Sep 11 2007
Number of rooted triangulations of type [n, 0] (see Brown paper eq (4.8)). - Michel Marcus, Jun 23 2013
Equivalently, number of rooted bridgeless planar maps with n edges. - Noam Zeilberger, Oct 06 2016
The September 2018 talk by Noam Zeilberger (see link to video) connects three topics (planar maps, Tamari lattices, lambda calculus) and eight sequences: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827. - N. J. A. Sloane, Sep 17 2018
Number of uniquely sorted permutations of [2n+1] that avoid the pattern 231. Also the number of uniquely sorted permutations of [2n+1] that avoid 132. - Colin Defant, Jun 13 2019
The sequence 1,3,13,68,... appears naturally in integral geometry, namely in the algebra of unitarily invariant valuations on complex space forms. - Andreas Bernig, Feb 02 2020

			G.f. = 1 + x + 3*x^2 + 13*x^3 + 68*x^4 + 399*x^5 + 2530*x^6 + 16965*x^7 + ...

C. F. Earl and L. J. March, Architectural applications of graph theory, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979.
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 714.
Handbook of Combinatorics, North-Holland '95, p. 891.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
W. T. Tutte, The enumerative theory of planar maps, in A Survey of Combinatorial Theory (J. N. Srivastava et al. eds.), pp. 437-448, North-Holland, Amsterdam, 1973.

T. D. Noe, Table of n, a(n) for n = 0..200
E. A. Bender and N. C. Wormald, The number of loopless planar maps, Discr. Math. 54:2 (1985), 235-237.
Andreas Bernig, Unitarily invariant valuations and Tutte's sequence, arXiv:2001.03372 [math.DG], 2020.
Alin Bostan, Frédéric Chyzak, Bérénice Delcroix-Oger, Guillaume Laplante-Anfossi, Vincent Pilaud, and Kurt Stoeckl, Diagonals of permutahedra and associahedra, Sém. Lotharingien Comb., 37th Formal Power Series Alg. Comb. (FPSAC 2025). See pp. 10-11.
Alin Bostan, Frédéric Chyzak, and Vincent Pilaud, Refined product formulas for Tamari intervals, arXiv:2303.10986 [math.CO], 2023.
T. Daniel Brennan, Christian Ferko, and Savdeep Sethi, A Non-Abelian Analogue of DBI from $T \overline{T}$, arXiv:1912.12389 [hep-th], 2019. See also SciPost Phys. (2020) Vol. 8, 052.
Enrica Duchi and Corentin Henriet, A bijection between Tamari intervals and extended fighting fish, arXiv:2206.04375 [math.CO], 2022.
William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.
Frédéric Chapoton, Sur le nombre d'intervalles dans les treillis de Tamari, arXiv:math/0602368 [math.CO], 2006.
Grégory Chatel, Vincent Pilaud, and Viviane Pons, The weak order on integer posets, arXiv:1701.07995 [math.CO], 2017.
Grégory Chatel and Viviane Pons, Counting smaller elements in the Tamari and m-Tamari lattices, arXiv preprint arXiv:1311.3922 [math.CO], 2013.
François David, A model of random surfaces with nontrivial critical behavior, Nuclear Physics B, v. 257 (1985), 543-576.
Colin Defant, Catalan Intervals and Uniquely Sorted Permutations, arXiv:1904.02627 [math.CO], 2019.
C. F. Earl and L. J. March, Architectural applications of graph theory, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979. (Annotated scanned copy)
Wenjie Fang, Planar triangulations, bridgeless planar maps and Tamari intervals, arXiv:1611.07922 [math.CO], 2016.
Joël Gay and Vincent Pilaud, The weak order on Weyl posets, arXiv:1804.06572 [math.CO], 2018.
C. Germain and J. Pallo, The number of coverings in four Catalan lattices, Intern. J. Computer Math., Vol. 61 (1996) pp. 19-28. (See p. 27.)
Hsien-Kuei Hwang, Mihyun Kang, and Guan-Huei Duh, Asymptotic Expansions for Sub-Critical Lagrangean Forms, LIPIcs Proceedings of Analysis of Algorithms 2018, Vol. 110. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2018.
Zhaoxiang Li and Yanpei Liu, Chromatic sums of general maps on the sphere and the projective plane, Discr. Math. 307 (2007), 78-87.
Michaël Moortgat, The Tamari order for D^3 and derivability in semi-associative Lambek-Grishin Calculus, 15th Workshop: Computational Logic and Applications (CLA 2020).
K. A. Penson, K. Górska, A. Horzela, and G. H. E. Duchamp, Hausdorff moment problem for combinatorial numbers of Brown and Tutte: exact solution, arXiv:2209.06574 [math.CO], 2022.
Viviane Pons, Combinatorics of the Permutahedra, Associahedra, and Friends, arXiv:2310.12687 [math.CO], 2023. See p. 37.
W. T. Tutte, A census of Hamiltonian polygons, Canad. J. Math., 14 (1962), 402-417.
William T. Tutte, A census of planar triangulations, Canad. J. Math. 14 (1962), 21-38. See Eq. 5.12.
William T. Tutte, On the enumeration of convex polyhedra, J. Combin. Theory Ser. B 28 (1980), no. 2, 105-126. MR0572468 (81j:05073).
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus. III: Nonseparable maps, J. Combinatorial Theory Ser. B 18 (1975), 222-259, eq. (6).
Noam Zeilberger, A sequent calculus for the Tamari order, arXiv:1701.02917 [cs.LO], 2017.
Noam Zeilberger, A Sequent Calculus for a Semi-Associative Law, arXiv preprint 1803.10030 [math.LO], 2018-2019 (A revised version of a 2017 conference paper).
Noam Zeilberger, A theory of linear typings as flows on 3-valent graphs, arXiv:1804.10540 [cs.LO], 2018.
Noam Zeilberger, A proof-theoretic analysis of the rotation lattice of binary trees, Part 1 (video), Rutgers Experimental Math Seminar, Sep 13 2018. See also Part 2.

Row sums of A342981.
Column 0 of A146305 and A341856; Column 2 of A255918.
Sequences mentioned in the Noam Zeilberger 2018 video: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827.
Cf. A000256, A006013, A027836, A069271, A263191, A290326.

[Binomial(4*n+1, n+1)-9*Binomial(4*n+1, n-1): n in [0..25]]; // Vincenzo Librandi, Nov 24 2016

A000260 := proc(n)
    2*(4*n+1)!/((n+1)!*(3*n+2)!) ;
end proc:

Table[Binomial[4n+1,n+1]-9*Binomial[4n+1,n-1],{n,0,25}] (* Harvey P. Dale, Aug 23 2011 *)
a[ n_] := SeriesCoefficient[ HypergeometricPFQ[ {1/2, 3/4, 1, 5/4}, {4/3, 5/3, 2}, 256/27 x], {x, 0, n}]; (* Michael Somos, Dec 23 2014 *)
terms = 22; G[] = 0; Do[G[x] = 1+x*G[x]^4 + O[x]^terms, terms];
CoefficientList[(2-G[x])*G[x]^2, x] (* Jean-François Alcover, Jan 13 2018, after Mark van Hoeij *)

{a(n) = if( n<0, 0, 2 * (4*n + 1)! / ((n + 1)! * (3*n + 2)!))}; /* Michael Somos, Sep 07 2005 */

{a(n) = binomial( 4*n + 2, n + 1) / ((2*n + 1) * (3*n + 2))}; /* Michael Somos, Mar 28 2012 */

def a(n):
    n = ZZ(n)
    return (4*n + 2).binomial(n + 1) // ((2*n + 1) * (3*n + 2))
# F. Chapoton, Aug 06 2015

a(n) = 2*(4*n+1)! / ((n+1)!*(3*n+2)!) = binomial(4*n+1, n+1) - 9*binomial(4*n+1, n-1).
G.f.: (2-g)*g^2 where g = 1 + x*g^4 is the g.f. of A002293. - Mark van Hoeij, Nov 10 2011
G.f.: hypergeom([1,1/2,3/4,5/4],[2,4/3,5/3],256*x/27) = 1 + 120*x/(Q(0)-120*x); Q(k) = 8*x*(2*k+1)*(4*k+3)*(4*k+5) + 3*(k+2)*(3*k+4)*(3*k+5) - 24*x*(k+2)*(2*k+3)*(3*k+4)*(3*k+5)*(4*k+7)*(4*k+9)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 25 2011
a(n) = binomial(4*n + 2, n + 1) / ((2*n + 1) * (3*n + 2)). - Michael Somos, Mar 28 2012
a(n) * (n+1) = A069271(n). - Michael Somos, Mar 28 2012
0 = F(a(n), a(n+1), ..., a(n+8)) for all n in Z where a(-1) = 3/4 and F() is a polynomial of degree 2 with integer coefficients and 29 monomials. - Michael Somos, Dec 23 2014
D-finite with recurrence: 3*(3*n+2)*(3*n+1)*(n+1)*a(n) - 8*(4*n+1)*(2*n-1)*(4*n-1)*a(n-1) = 0. - R. J. Mathar, Oct 21 2015
a(n) = Sum_{k=1..A000108(n)} k * A263191(n,k). - Alois P. Heinz, Nov 16 2015
a(n) ~ 2^(8*n+7/2) / (sqrt(Pi) * n^(5/2) * 3^(3*n+5/2)). - Vaclav Kotesovec, Feb 26 2016
E.g.f.: 3F3(1/2,3/4,5/4; 4/3,5/3,2; 256*x/27). - Ilya Gutkovskiy, Feb 01 2017
From Gheorghe Coserea, Aug 17 2017: (Start)
G.f. y(x) satisfies:
0 = x^3*y^4 + 3*x^2*y^3 + x*(8*x+3)*y^2 - (20*x-1)*y + 16*x-1.
0 = x*(256*x - 27)*deriv(y,x) - 8*x^2*y^3 - 25*x*y^2 + 4*(24*x-11)*y + 44.
(End)
From Karol A. Penson, Apr 06 2022: (Start)
a(n) = Integral_{x=0...256/27} x^n*W(x), where
W(x) = (sqrt(2)/Pi)*(h1(x) - h2(x) + h3(x)) and
h1(x) = 3F2([-6/12,-2/12,  2/12], [ 3/12,  9/12], (27*x)/256)/((x/2)^(1/2));
h2(x) = 3F2([-3/12, 1/12,  5/12], [ 6/12, 15/12], (27*x)/256)/(x^(1/4));
h3(x) = 3F2([ 3/12, 7/12, 11/12], [18/12, 21/12], (27*x)/256)/(x^(-1/4)*32).
This integral representation is unique as the solution of n-th Hausdorff power moment of the function W. Using only the definition of a(n), W(x) can be proven to be positive. W(x) is singular at x = 0 and for x > 0 is monotonically decreasing to zero at x = 256/27. (End)
a(n) = (1/27^n) * Product_{1 <= i <= j <= 3*n} (3*i + j + 3)/(3*i + j - 1). Cf. A006013. - Peter Bala, Feb 21 2023

Edited by F. Chapoton, Feb 03 2011

A000168 a(n) = 23^n(2n)!/(n!(n+2)!).

Original entry on oeis.org

1, 2, 9, 54, 378, 2916, 24057, 208494, 1876446, 17399772, 165297834, 1602117468, 15792300756, 157923007560, 1598970451545, 16365932856990, 169114639522230, 1762352559231660, 18504701871932430, 195621134074714260, 2080697516976506220, 22254416920705240440, 239234981897581334730, 2583737804493878415084

Offset: 0

N. J. A. Sloane

Number of rooted planar maps with n edges. - Don Knuth, Nov 24 2013
Number of rooted 4-regular planar maps with n vertices.
Also, number of doodles with n crossings, irrespective of the number of loops.
From Karol A. Penson, Sep 02 2010: (Start)
Integral representation as n-th moment of a positive function on the (0,12) segment of the x axis. This representation is unique as it is the solution of the Hausdorff moment problem.
a(n) = Integral_{x=0..12} ((x^n*(4/9)*(1 - x/12)^(3/2)) / (Pi*sqrt(x/3))).  (End)
Also, the number of distinct underlying shapes of closed normal linear lambda terms of a given size, where the shape of a lambda term abstracts away from its variable binding. [N. Zeilberger, 2015] - N. J. A. Sloane, Sep 18 2016
The September 2018 talk by Noam Zeilberger (see link to video) connects three topics (planar maps, Tamari lattices, lambda calculus) and eight sequences: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827. - N. J. A. Sloane, Sep 17 2018
Number of well-labeled trees (Bona, 2015). - N. J. A. Sloane, Dec 25 2018

			G.f. = 1 + 2*x + 9*x^2 + 54*x^3 + 378*x^4 + 2916*x^5 + 24057*x^6 + 208494*x^7 + ...

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 319, 353.
E. R. Canfield, Calculating the number of rooted maps on a surface, Congr. Numerantium, 76 (1990), 21-34.
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 714.
V. A. Liskovets, A census of nonisomorphic planar maps, in Algebraic Methods in Graph Theory, Vol. II, ed. L. Lovasz and V. T. Sos, North-Holland, 1981.
V. A. Liskovets, Enumeration of nonisomorphic planar maps, Selecta Math. Sovietica, 4 (No. 4, 1985), 303-323.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..925 [Terms 0 to 100 computed by T. D. Noe; terms 101 to 925 by G. C. Greubel, Jan 15 2017]
Marie Albenque and Dominique Poulalhon, A Generic Method for Bijections between Blossoming Trees and Planar Maps, Electron. J. Combin., 22 (2015), #P2.38.
J.-L. Baril, R. Genestier, A. Giorgetti, and A. Petrossian, Rooted planar maps modulo some patterns, Preprint 2016.
Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, A family of Bell transformations, arXiv:1803.07727 [math.CO], 2018.
Valentin Bonzom, Guillaume Chapuy, Maciej Dolega, Enumeration of non-oriented maps via integrability, Alg. Combin. 5 (6) (2022) p 1363-1390, A.1.
M. Bousquet-Mélou, Limit laws for embedded trees, arXiv:math/0501266 [math.CO], 2005.
M. Bousquet-Mélou and A. Jehanne, Polynomial equations with one catalytic variable, algebraic series and map enumeration, arXiv:math/0504018 [math.CO], 2005.
Sean R. Carrell and Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], 2014.
R. Cori and B. Vauquelin, Planar maps are well labeled trees, Canad. J. Math., 33 (1981), 1023-1042.
S. R. Finch, An exceptional convolutional recurrence, arXiv:2408.12440 [math.CO], 22 Aug 2024.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 516
A. Giorgetti, R. Genestier, and V. Senni, Software Engineering and Enumerative Combinatorics, slides from a talk at MAP 2014.
Hsien-Kuei Hwang, Mihyun Kang, and Guan-Huei Duh, Asymptotic Expansions for Sub-Critical Lagrangean Forms, LIPIcs Proceedings of Analysis of Algorithms 2018, Vol. 110. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2018.
C. Kassel, On combinatorial zeta functions, Slides from a talk, Potsdam, 2015.
Sergey Kitaev, Anna de Mier, and Marc Noy, On the number of self-dual rooted maps, European J. Combin. 35 (2014), 377--387. MR3090510. See Eq. (1). - _N. J. A. Sloane_, May 19 2014
Evgeniy Krasko and Alexander Omelchenko, Enumeration of r-regular maps on the torus. Part I: Rooted maps on the torus, the projective plane and the Klein bottle. Sensed maps on the torus, Discrete Mathematics (2019) Vol. 342, Issue 2, 584-599. Also arXiv:1709.03225 [math.CO].
V. A. Liskovets, Enumeration of nonisomorphic planar maps, Journal of Graph Theory, Volume 5, Issue 1, pages 115-117, Spring 1981.
Valery A. Liskovets, A reductive technique for enumerating non-isomorphic planar maps, Discrete Math. 156 (1996), no. 1-3, 197--217. MR1405018 (97f:05087) - From _N. J. A. Sloane_, Jun 03 2012
R. C. Mullin, On the average activity of a spanning tree of a rooted map, J. Combin. Theory, 3 (1967), 103-121.
R. C. Mullin, On the average activity of a spanning tree of a rooted map, J. Combin. Theory, 3 (1967), 103-121. [Annotated scanned copy]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, Une méthode pour obtenir la fonction génératrice d'une série, arXiv:0912.0072 [math.NT], 2009; FPSAC 1993, Florence. Formal Power Series and Algebraic Combinatorics.
C. Reutenauer and M. Robado, On an algebraicity theorem of Kontsevich, FPSAC 2012, Nagoya, Japan DMTCS proc. AR, 2012, 241-248. - From _N. J. A. Sloane_, Dec 23 2012
G. Schaeffer and P. Zinn-Justin, On the asymptotic number of plane curves and alternating knots, arXiv:math-ph/0304034, 2003-2004.
W. T. Tutte, A Census of Planar Maps, Canad. J. Math. 15 (1963), 249-271.
Noam Zeilberger, Counting isomorphism classes of beta-normal linear lambda terms, arXiv:1509.07596 [cs.LO], 2015.
Noam Zeilberger, Towards a mathematical science of programming, Preprint 2015.
Noam Zeilberger, Linear lambda terms as invariants of rooted trivalent maps, arXiv preprint arXiv:1512.06751 [cs.LO], 2015.
Noam Zeilberger, A theory of linear typings as flows on 3-valent graphs, arXiv:1804.10540 [cs.LO], 2018.
Noam Zeilberger, A Sequent Calculus for a Semi-Associative Law, arXiv preprint 1803.10030, March 2018 (A revised version of a 2017 conference paper)
Noam Zeilberger, A proof-theoretic analysis of the rotation lattice of binary trees, Part 1 (video), Part 2, Rutgers Experimental Math Seminar, Sep 13 2018.
Noam Zeilberger and Alain Giorgetti, A correspondence between rooted planar maps and normal planar lambda terms, arXiv:1408.5028 [cs.LO], 2014-2015; Logical Methods in Computer Science, vol. 11 (3:22), 2015, pp. 1-39.
Jian Zhou, Fat and Thin Emergent Geometries of Hermitian One-Matrix Models, arXiv:1810.03883 [math-ph], 2018.

Sequences mentioned in the Noam Zeilberger 2018 video: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827.
First row of array A101486.
Cf. A005470.
Rooted maps with n edges of genus g for 0 <= g <= 10: this sequence, A006300, A006301, A104742, A215402, A238355, A238356, A238357, A238358, A238359, A238360.

[(2*Catalan(n)*3^n)/(n+2): n in [1..30]]; // Vincenzo Librandi, Sep 04 2014

Maple
```
A000168:=n->2*3^n*(2*n)!/(n!*(n+2)!);
```

Table[(2*3^n*(2n)!)/(n!(n+2)!),{n,0,20}] (* Harvey P. Dale, Jul 25 2011 *)
a[ n_] := If[ n < 0, 0, 2 3^n (2 n)!/(n! (n + 2)!)] (* Michael Somos, Nov 25 2013 *)
a[ n_] := SeriesCoefficient[ Hypergeometric2F1[ 1/2, 1, 3, 12 x], {x, 0, n}] (* Michael Somos, Nov 25 2013 *)

{a(n) = if( n<0, 0, 2 * 3^n * (2*n)! / (n! * (n+2)!))}; /* Michael Somos, Nov 25 2013 */

G.f. A(z) satisfies A(z) = 1 - 16*z + 18*z*A(z) - 27*z^2*A(z)^2.
G.f.: F(1/2,1;3;12x). - Paul Barry, Feb 04 2009
a(n) = 2*3^n*A000108(n)/(n+2). - Paul Barry, Feb 04 2009
D-finite with recurrence: (n + 1) a(n) = (12 n - 18) a(n - 1). - Simon Plouffe, Feb 09 2012
G.f.: 1/54*(-1+18*x+(-(12*x-1)^3)^(1/2))/x^2. - Simon Plouffe, Feb 09 2012
0 = a(n)*(+144*a(n+1) - 42*a(n+2)) + a(n+1)*(+18*a(n+1) + a(n+2)) if n>=0. - Michael Somos, Jan 31 2014
a(n) ~ 2*(12^n)/((n^2+3*n)*sqrt(Pi*n)). - Peter Luschny, Nov 25 2015
E.g.f.: exp(6*x)*(12*x*BesselI(0,6*x) - (1 + 12*x)*BesselI(1,6*x))/(9*x). - Ilya Gutkovskiy, Feb 01 2017
From Amiram Eldar, Jan 08 2023: (Start)
Sum_{n>=0} 1/a(n) = 1887/1331 + 3240*arccosec(2*sqrt(3))/(1331*sqrt(11)).
Sum_{n>=0} (-1)^n/a(n) = 1563/2197 - 3240*arccosech(2*sqrt(3))/(2197*sqrt(13)). (End)

More terms from Joerg Arndt, Feb 26 2014

A000139 a(n) = 2(3n)! / ((2n+1)!(n+1)!).

Original entry on oeis.org

2, 1, 2, 6, 22, 91, 408, 1938, 9614, 49335, 260130, 1402440, 7702632, 42975796, 243035536, 1390594458, 8038677054, 46892282815, 275750636070, 1633292229030, 9737153323590, 58392041019795, 352044769046880, 2132866978427640, 12980019040145352, 79319075627675556

Offset: 0

N. J. A. Sloane, entry revised Apr 24 2012

This sequence arises in many different contexts, and over the years it has had several different definitions. I have now changed the definition back to one of the earlier ones, a self-contained formula. - N. J. A. Sloane, Apr 24 2012
The number of 2-stack sortable permutations on n letters (n >= 1).
The number of rooted non-separable planar maps with n+1 edges. - Valery A. Liskovets, Mar 17 2005
The shifted sequence starting with a(1): Number of quadrangular dissections of a square, counted by the number of vertices. Rooted, non-separable planar maps with no multiple edges, in which each non-root face has degree 4.
Number of left ternary trees having n nodes (n>=1). - Emeric Deutsch, Jul 23 2006
A combinatorial interpretation for this sequence in terms of a family of plane trees is given in [Schaeffer, Corollary 2 with k = 3]. - Peter Bala, Oct 12 2011
Number of canopy intervals in the Tamari lattices, see [Préville-Ratelle and Viennot, section 6]. - F. Chapoton, Apr 19 2015
The number of fighting fish (branching polyominoes). - David Bevan, Jan 10 2018
The number of 1324-avoiding dominoes (gridded permutations). - David Bevan, Jan 10 2018
For n > 0, a(n) is the number of simple strong triangulations of a fixed quadrilateral with n interior nodes. See A210664. - Andrew Howroyd, Feb 24 2021
Conjecture: a(n) is odd iff n is a term of A022341. - Peter Bala, Jul 24 2025

			G.f. = 2 + x + 2*x^2 + 6*x^3 + 22*x^4 + 91*x^5 + 408*x^6 + 1938*x^7 + ...

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 365.
Eric S. Egge, Defying God: The Stanley-Wilf Conjecture, Stanley-Wilf Limits, and a Two-Generation Explosion of Combinatorics, pp. 65-82 of "A Century of Advancing Mathematics", ed. S. F. Kennedy et al., MAA Press 2015.
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 714.
S. Kitaev, Patterns in Permutations and Words, Springer-Verlag, 2011. See p. 399 Table A.7
W. F. Lunnon, Counting polyominoes, pp. 347-372 of A. O. L. Atkin and B. J. Birch, editors, Computers in Number Theory. Academic Press, NY, 1971.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.41.

T. D. Noe, Table of n, a(n) for n = 0..200
A. M. Baxter, Algorithms for Permutation Statistics, Ph. D. Dissertation, Rutgers University, May 2011. See p. 15.
E. Ben-Naim and P. L. Krapivsky, Popularity-Driven Networking, arXiv preprint arXiv:1112.0049 [cond-mat.stat-mech], 2011.
Alin Bostan, Frédéric Chyzak, Bérénice Delcroix-Oger, Guillaume Laplante-Anfossi, Vincent Pilaud, and Kurt Stoeckl, Diagonals of permutahedra and associahedra, Sém. Lotharingien Comb., 37th Formal Power Series Alg. Comb. (FPSAC 2025). See pp. 10-11.
David Bevan, Robert Brignall, Andrew Elvey Price and Jay Pantone, A structural characterisation of Av(1324) and new bounds on its growth rate, arXiv preprint arXiv:1711.10325 [math.CO], 2017.
Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, A family of Bell transformations, arXiv:1803.07727 [math.CO], 2018.
Alin Bostan, Frédéric Chyzak, and Vincent Pilaud, Refined product formulas for Tamari intervals, arXiv:2303.10986 [math.CO], 2023.
M. Bousquet-Mélou and A. Jehanne, Polynomial equations with one catalytic variable, algebraic series and map enumeration, arXiv:math/0504018 [math.CO], 2005; J Comb. Thy. B 96 (2006), 623-672.
W. G. Brown, Enumeration of non-separable planar maps, Canad. J. Math., 15 (1963), 526-545.
Colin Defant, Counting 3-Stack-Sortable Permutations, arXiv:1903.09138 [math.CO], 2019.
A. Del Lungo, F. Del Ristoro and J.-G. Penaud, Left ternary trees and non-separable rooted planar maps, Theor. Comp. Sci., 233, 2000, 201-215.
E. Duchi, V. Guerrini, S. Rinaldi and G. Schaeffer, Fighting fish: enumerative properties, Sém. Lothar. Combin. 78B (2017), Art. 43, 12 pp.
E. Duchi, V. Guerrini, S. Rinaldi, and G. Schaeffer, Fighting fish, J. Phys. A, Math. Theor. 50, No. 2, Article ID 024002, 16 p. (2017).
Enrica Duchi and Corentin Henriet, A bijection between rooted planar maps and generalized fighting fish, arXiv:2210.16635 [math.CO], 2022.
S. Dulucq, S. Gire, and O. Guibert, A combinatorial proof of J. West's conjecture Discrete Math. 187 (1998), no. 1-3, 71--96. MR1630680(99f:05053).
S. Dulucq, S. Gire, and J. West, Permutations with forbidden subsequences and nonseparable planar maps, Proceedings of the 5th Conference on Formal Power Series and Algebraic Combinatorics (Florence, 1993). Discrete Math. 153 (1996), no. 1-3, 85-103. MR1394948 (98a:05081)
Eric S. Egge, Defying God: The Stanley-Wilf Conjecture, Stanley-Wilf Limits, and a Two-Generation Explosion of Combinatorics, MAA Focus, August/September 2015, pp. 33-34. [Annotated scanned copy]
W. Fang, Fighting fish and two-stack sortable permutations, arXiv preprint arXiv:1711.05713 [math.CO], 2017.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 713
I. Gessel and G. Xin, The generating function of ternary trees and continued fractions, Electron. J Combin. 13 (2006), paper 53.
Juan B. Gil, Oscar A. Lopez, and Michael D. Weiner, A positional statistic for 1324-avoiding permutations, arXiv:2311.18227 [math.CO], 2023.
O. Guibert, Stack words, standard Young tableaux, permutations with forbidden subsequences and planar maps, Discr. Math., 210 (2000), 71-85.
Elizabeth Hartung, Hung Phuc Hoang, Torsten Mütze, and Aaron Williams, Combinatorial generation via permutation languages. I. Fundamentals, arXiv:1906.06069 [cs.DM], 2019.
Hsien-Kuei Hwang, Mihyun Kang, and Guan-Huei Duh, Asymptotic Expansions for Sub-Critical Lagrangean Forms, LIPIcs Proceedings of Analysis of Algorithms 2018, Vol. 110. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2018.
S. Kitaev, P. Salimov, C. Severs and H. Ulfarsson, Restricted non-separable planar maps and some pattern avoiding permutations, 2012.
S. Kitaev, P. Salimov, C. Severs and H. Ulfarsson, Restricted non-separable planar maps and some pattern avoiding permutations, Discrete Applied Mathematics, Volume 161, Issues 16-17, November 2013, Pages 2514-2526.
Sergey Kitaev, Anna de Mier, and Marc Noy, On the number of self-dual rooted maps, European J. Combin. 35 (2014), 377-387. MR3090510. See Theorem 1.
Sergey Kitaev, Pavel Salimov, Christopher Severs, and Henning Ulfarsson, Restricted non-separable planar maps and some pattern avoiding permutations, arXiv preprint arXiv:1202.1790 [math.CO], 2012. [Original title: Restricted rooted non-separable planar maps]
Arturo Merino and Torsten Mütze, Combinatorial generation via permutation languages. III. Rectangulations, arXiv:2103.09333 [math.CO], 2021.
Alois Panholzer, Parking function varieties for combinatorial tree models, arXiv:2007.14676 [math.CO], 2020.
Permutation Pattern Avoidance Library (PermPAL), 1324 Domino
L.-F. Préville-Ratelle and X. Viennot, An extension of Tamari lattices, arXiv preprint arXiv:1406.3787 [math.CO], 2014.
G. Schaeffer, A combinatorial interpretation of super-Catalan numbers of order two, (2001).
W. T. Tutte, A Census of Planar Maps, Canad. J. Math. 15 (1963), 249-271.
J. West, Sorting twice through a stack, Conference on Formal Power Series and Algebraic Combinatorics (Bordeaux, 1991), Theoret. Comput. Sci. 117 (1993), no. 1-2, 303-313.
D. Zeilberger, A proof of Julian West's conjecture that the number of two-stacksortable permutations of length n is 2(3n)!/((n + 1)!(2n + 1)!), Discrete Math., 102 (1992), 85-93.
P. Zinn-Justin and J.-B. Zuber, Matrix integrals and the counting of tangles and links, Discrete Math 246 (2002), 343-360.
P. Zinn-Justin and J.-B. Zuber, Matrix integrals and the generation and counting of virtual tangles and links, arXiv:math-ph/0303049, 2003; J. Knot Theor. Ramifications 13 (2004) 325-356.

Cf. A000142, A000309, A001764, A005802, A006335, A004677, A121545, A218473, A233389.

Row m=1 of A210664(n>0).

a000139 0 = 2
a000139 n = ((3 * n) `a007318` (2 * n + 1)) `div` a000217 n
-- Reinhard Zumkeller, Feb 17 2013

[2*Factorial(3*n)/(Factorial(2*n+1)*Factorial(n+1)): n in [0..25]]; // Vincenzo Librandi, Apr 20 2015

A000139 := n->2*(3*n)!/((2*n+1)!*((n+1)!)): seq(A000139(n), n=0..23);

Table[(2(3n)!)/((2n+1)!(n+1)!),{n,0,30}] (* Harvey P. Dale, Mar 31 2013 *)

a(n)=binomial(3*n,n)*2/((n+1)*(2*n+1)); \\ Joerg Arndt, Jul 21 2014

from sympy import binomial
def A000139(n): return (binomial(3*n, n)*2)//((n+1)*(2*n+1))

A000139_list = [2]
for n in range(1,30):
    A000139_list.append(3*(3*n-2)*(3*n-1)*A000139_list[-1]//(2*n+2)//(2*n+1)) # Chai Wah Wu, Apr 02 2021

def A000139(n): return (binomial(3*n, n)*2)//((n+1)*(2*n+1))
[A000139(n) for n in (0..23)]  # Peter Luschny, Jun 17 2013

a(n) = 2*binomial(3*n, 2*n+1)/(n*(n+1)), or 2*(3*n)!/((2*n+1)!*((n+1)!)).
Using Stirling's formula in A000142 it is easy to get the asymptotic expression a(n) ~ (27/4)^n / (sqrt(Pi*n / 3) * (2*n + 1) * (n + 1)). - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 13 2001
G.f.: A(z) = 2 + z*B(z), where B(z) = 1 - 8*z + 2*z*(5-6*z)*B - 2*z^2*(1+3*z)*B^2 - z^4*B^3.
G.f.: (2/(3*x)) * (hypergeom([-2/3, -1/3],[1/2],(27/4)*x)-1). - Mark van Hoeij, Nov 02 2009
G.f.: (2-3*R)/(R-1)^2 where R := RootOf(x-t*(t-1)^2,t) is an algebraic function in Maple notation. - Mark van Hoeij, Nov 08 2011
G.f.: 2*Q(0), where Q(k) = 1 + 3*x*(3*k+1)*(6*k+1)/(2*(k+1)*(4*k+3) - 6*x*(k+1)*(3*k+2)*(4*k+3)*(6*k+5)/(3*x*(3*k+2)*(6*k+5) + (2*k+3)*(4*k+5)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 30 2013
E.g.f.: 2*Q(0), where Q(k) = 1 + 3*x*(3*k+1)*(6*k+1)/(2*(k+1)*(2*k+1)*(4*k+3) - 6*x*(k+1)*(2*k+1)*(3*k+2)*(4*k+3)*(6*k+5)/(3*x*(3*k+2)*(6*k+5) + (2*k+2)*(2*k+3)*(4*k+5)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 30 2013
a(n) = A007318(3*n, 2*n+1)/A000217(n) for n > 0. - Reinhard Zumkeller, Feb 17 2013
a(n) is the n-th Hausdorff moment of the positive function w(x) defined on (0,27) which is equal to w(x) = 3*sqrt(3)*2^(2/3)*(3-sqrt(81-12*x)/9)*(1+sqrt(81-12*x)/9)^(1/3)/(8*Pi*x^(2/3))-sqrt(3)*2^(1/3)*(3+sqrt(81-12*x)/9)*(1+sqrt(81-12*x)/9)^(-1/3)/(4*Pi*x^(1/3)), that is, a(n) is the integral Integral_{x=0..27/4} x^n*w(x) dx, n >= 0. The function w(x) is unique. - Karol A. Penson, Jun 17 2013
D-finite with recurrence 2*(n+1)*(2*n+1)*a(n) -3*(3*n-1)*(3*n-2)*a(n-1)=0. - R. J. Mathar, Aug 21 2014
G.f. A(z) is related to the g.f. M(z) of A000168 by M(z) = 1 + A(z*M(z)^2) (see Tutte 1963, equation 6.3). - Noam Zeilberger, Nov 02 2016
From Ilya Gutkovskiy, Jan 17 2017: (Start)
E.g.f.: 2*2F2(1/3,2/3; 3/2,2; 27*x/4).
Sum_{n>=0} 1/a(n) = (1/2)*3F2(1,3/2,2; 1/3,2/3; 4/27) = 2.226206199291261... (End)
G.f. A(z) is the solution to the initial value problem 4*A + 2*z*A' = 8 + 3*z*A + 9*z^2*A' + 2*z^2*A*A', A(0) = 2. - Bjarki Ágúst Guðmundsson, Jul 03 2017
a(n+1) = a(n)*3*(3*n+1)*(3*n+2)/(2*(n+2)*(2*n+3)). - Chai Wah Wu, Apr 02 2021
a(n) = 4*(3*n)!/(n!*(2*n+2)!). - Chai Wah Wu, Dec 15 2021
From Peter Bala, Feb 05 2022: (Start)
O.g.f.: A(x) = T(x)*(3 - T(x)), where T(x) = 1 + x*T(x)^3 is the o.g.f. of A001764.
(1/x)*(A(x) - 2)/(A(x) - 1) = 1 + x + 3*x^2 + 11*x^3 + 46*x^4 + 209*x^5 + ... is the o.g.f. of A233389.
1 + 2*x*A'(2*x)/A(2*x) = 1 + x + 7*x^2 + 61*x^3 + 591*x^4 + 6101*x^6 + ... is the o.g.f. of A218473.
Let B(x) = 1 + x*(A(x) - 1). Then x*B'(x)/B(x) = x + x^2 + 4*x^3 + 17*x^4 + 81*x^5 + ... is the o.g.f. of A121545. (End)

A062980 a(0) = 1, a(1) = 5; for n > 1, a(n) = 6na(n-1) + Sum_{k=1..n-2} a(k)*a(n-k-1).

Original entry on oeis.org

1, 5, 60, 1105, 27120, 828250, 30220800, 1282031525, 61999046400, 3366961243750, 202903221120000, 13437880555850250, 970217083619328000, 75849500508999712500, 6383483988812390400000, 575440151532675686278125, 55318762960656722780160000

Offset: 0

Michael Praehofer (praehofer(AT)ma.tum.de), Jul 24 2001

Number of rooted unlabeled connected triangular maps on a compact closed oriented surface with 2n faces (and thus 3n edges). [Vidal]
Equivalently, the number of pair of permutations (sigma,tau) up to simultaneous conjugacy on a pointed set of size 6*n with sigma^3=tau^2=1, acting transitively and with no fixed point. [Vidal]
Also, the asymptotic expansion of the Airy function Ai'(x)/Ai(x) = -sqrt(x) - 1/(4x) + Sum_{n>=2} (-1)^n a(n) (4x)^ (1/2-3n/2). [Praehofer]
Maple 6 gives the wrong asymptotics of Ai'(x)=AiryAi(1,x) as x->oo apart from the 3rd term. Therefore asympt(AiryAi(1,x/4)/AiryAi(x/4),x); reproduces only the value a(1)=1 correctly.
Number of closed linear lambda terms (see [Bodini, Gardy, Jacquot, 2013] and [N. Zeilberger, 2015] references). - Pierre Lescanne, Feb 26 2017
Proved (bijection) by O. Bodini, D. Gardy, A. Jacquot (2013). - Olivier Bodini, Mar 30 2018
The September 2018 talk by Noam Zeilberger (see link to video) connects three topics (planar maps, Tamari lattices, lambda calculus) and eight sequences: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827. - N. J. A. Sloane, Sep 17 2018

			1 + 5*x + 60*x^2 + 1105*x^3 + 27120*x^4 + 828250*x^5 + 30220800*x^6 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..100
O. Bodini, D. Gardy, and A. Jacquot, Asymptotics and random sampling for BCI and BCK lambda terms, Theor. Comput. Sci. 502: 227-238 (2013).
Jørgen Ellegaard Andersen, Gaëtan Borot, Leonid O. Chekhov and Nicolas Orantin, The ABCD of topological recursion, arXiv:1703.03307 [math-ph], 2017. [See Appendix A.1]
Laura Ciobanu and Alexander Kolpakov, Free subgroups of free products and combinatorial hypermaps, arXiv:1708.03842 [math.CO], 2017.
P. Cvitanovic, B. Lautrup and R. B. Pearson, The number and weights of Feynman diagrams, Phys. Rev. D18, (1978), 1939-1949, (eq 3.14 and fig 1b). DOI:10.1103/PhysRevD.18.1939
Bertrand Eynard, Topological expansion for the 1-hermitian matrix model correlation functions, Journal of High Energy Physics 11 (2004). [See Eq. (5.12) and Appendix A]
Steven R. Finch, Shapes of binary trees, June 24, 2004. [Cached copy, with permission of the author]
Steven R. Finch, An exceptional convolutional recurrence, arXiv:2408.12440 [math.CO], 2024.
Éric Fusy, Luca Lionni, Adrian Tanasa, Combinatorial study of graphs arising from the Sachdev-Ye-Kitaev model, arXiv:1810.02146 [math.CO], 2018.
Katarzyna Grygiel, Pawel M. Idziak and Marek Zaionc, How big is BCI fragment of BCK logic, arXiv preprint arXiv:1112.0643 [cs.LO], 2011. [From _N. J. A. Sloane_, Feb 21 2012]
Kristian Gustavsson and Bernhard Mehlig, Statistical models for spatial patterns of heavy particles in turbulence, arxiv:1412.4374 [physics.flu-dyn], 2014-2016. [See Figure 14]
M. J. Kearny and R. J. Martin, Normalized functionals of first passage Brownian motion and a curious connection with the maximal relative height of fluctuating interfaces, J. Phys. A, Math. Theor. 49, No. 19, Article ID 195001, 20 p. (2016).
George Kaye, A visualiser for linear lambda-terms as 3-valent rooted maps, University of Birmingham (UK, 2019).
S. Janson, The Wiener index of simply generated random trees, Random Structures Algorithms 22 (2003) 337-358.
Michael J. Kearney, Satya N. Majumdar and Richard J. Martin, The first-passage area for drifted Brownian motion and the moments of the Airy distribution, arXiv:0706.2038 [cond-mat.stat-mech], 2007. [a(n) = 8^n * K_n from Eq. (3)]
Pierre Lescanne, Quantitative aspects of linear and affine closed lambda terms, arXiv:1702.03085 [cs.DM], 2017. Also in ACM Transactions on Computational Logic (TOCL 2019), Vol. 19, No. 2, Article No. 9.
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, Aequat. Math., 80 (2010), 291-318. See p. 292.
NIST Digital Library of Mathematical Functions, Airy Functions.
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.
Paul Tarau and Valeria de Paiva, Deriving Theorems in Implicational Linear Logic, Declaratively, arXiv:2009.10241 [cs.LO], 2020. See also Github, (2020).
Samuel Vidal and Michel Petitot, Counting Rooted and Unrooted Triangular Maps, Journal of Nonlinear Systems and Applications (2009) 151-154.
Eric Weisstein's World of Mathematics, Airy Functions, contains the definitions of Ai(x), Bi(x).
Noam Zeilberger, Counting isomorphism classes of beta-normal linear lambda terms, arXiv:1509.07596 [cs.LO], 2015.
Noam Zeilberger, Linear lambda terms as invariants of rooted trivalent maps, arXiv preprint arXiv:1512.06751 [cs.LO], 2015.
Noam Zeilberger, A theory of linear typings as flows on 3-valent graphs, arXiv:1804.10540 [cs.LO], 2018.
Noam Zeilberger, A Sequent Calculus for a Semi-Associative Law, arXiv preprint 1803.10030 [math.LO], March 2018 (A revised version of a 2017 conference paper).
Noam Zeilberger, A proof-theoretic analysis of the rotation lattice of binary trees, Part 1 (video), Part 2, Rutgers Experimental Math Seminar, Sep 13 2018.
Noam Zeilberger and Alain Giorgetti, A correspondence between rooted planar maps and normal planar lambda terms, arXiv:1408.5028 [cs.LO], 2014-2015; Logical Methods in Computer Science, vol. 11 (3:22), 2015, pp. 1-39.

Sequences mentioned in the Noam Zeilberger 2018 video: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827.
With interspersed zeros column 3 of A380622.
Pointed version of A129114.
Connected pointed version of A129115.
Cf. A060506, A060507, A094199, A121350, A121352, A005133, A172455.

a062980 n = a062980_list !! n
a062980_list = 1 : 5 : f 2 [5,1] where
   f u vs'@(v:vs) = w : f (u + 1) (w : vs') where
     w = 6 * u * v + sum (zipWith (*) vs_ $ reverse vs_)
     vs_ = init vs
-- Reinhard Zumkeller, Jun 03 2013

a:= proc(n) option remember; `if`(n<2, 4*n+1,
      6*n*a(n-1) +add(a(k)*a(n-k-1), k=1..n-2))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Mar 31 2017

max = 16; f[y_] := -Sqrt[x] - 1/(4*x) + Sum[(-1)^n*a[n]*(4*x)^(1/2 - 3*(n/2)), {n, 2, max}] /. x -> 1/y^2; s[y_] := Normal[ Series[ AiryAiPrime[x] / AiryAi[x], {x, Infinity, max + 7}]] /. x -> 1/y^2; sol = SolveAlways[ Simplify[ f[y] == s[y], y > 0], y] // First; Join[{1, 5}, Table[a[n], {n, 3, max}] /. sol] (* Jean-François Alcover, Oct 09 2012, from Airy function asymptotics *)
a[0] = 1; a[n_] := a[n] = (6*(n-1)+4)*a[n-1] + Sum[a[i]*a[n-i-1], {i, 0, n-1}]; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Nov 29 2013, after Vladimir Reshetnikov *)

{a(n) = local(A); n++; if( n<1, 0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (6*k - 8) * A[k-1] + sum( j=1, k-1, A[j] * A[k-j])); A[n])} /* Michael Somos, Jul 24 2011 */

from sympy.core.cache import cacheit
@cacheit
def a(n): return n*4 + 1 if n<2 else 6*n*a(n - 1) + sum(a(k)*a(n - k - 1) for k in range(1, n - 1))
print([a(n) for n in range(21)]) # Indranil Ghosh, Aug 09 2017

With offset 1, then a(1) = 1 and, for n > 1, a(n) = (6*n-8)*a(n-1) + Sum_{k=1..n-1} a(k)*a(n-k) [Praehofer] [Martin and Kearney].
a(n) = (6/Pi^2)*Integral_{x=0..oo} ((4*x)^(3*n/2)/(Ai(x)^2 + Bi(x)^2)) dt. - Vladimir Reshetnikov, Sep 24 2013
a(n) ~ 3 * 6^n * n! / Pi. - Vaclav Kotesovec, Jan 19 2015
0 = 6*x^2*y' + x*y^2 + (4*x-1)*y + 1, where y(x) = Sum_{n>=0} a(n)*x^n. - Gheorghe Coserea, Apr 02 2017
From Peter Bala, May 21 2017: (Start)
G.f. as an S-fraction: A(x) = 1/(1 - 5*x/(1 - 7*x/(1 - 11*x/(1 - 13*x/(1 - ... - (6*n - 1)*x/(1 - (6*n + 1)*x/(1 - .... See Stokes.
x*A(x) = B(x)/(1 + 2*B(x)), where B(x) = x + 7*x^2 + 84*x^3 + 1463*x^4 + ... is the o.g.f. of A172455.
A(x) = 1/(1 + 2*x - 7*x/(1 - 5*x/(1 - 13*x/(1 - 11*x/(1 - ... - (6*n + 1)*x/(1 - (6*n - 1)*x/(1 - .... (End)

Entry revised by N. J. A. Sloane based on comments from Samuel A. Vidal, Mar 30 2007

A267827 Number of closed indecomposable linear lambda terms with 2n+1 applications and abstractions.

Original entry on oeis.org

1, 2, 20, 352, 8624, 266784, 9896448, 426577920, 20918138624, 1149216540160, 69911382901760, 4665553152081920, 338942971881472000, 26631920159494995968, 2250690001888540950528, 203595258621775065120768, 19629810220331494121865216

Offset: 0

Noam Zeilberger, Jan 21 2016

nonn

A linear lambda term is indecomposable if it has no closed proper subterm.
Equivalently, number of closed bridgeless rooted trivalent maps (on compact oriented surfaces of arbitrary genus) with 2n+1 trivalent vertices (and 1 univalent vertex).
The September 2018 talk by Noam Zeilberger (see link to video) connects three topics (planar maps, Tamari lattices, lambda calculus) and eight sequences: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827. - N. J. A. Sloane, Sep 17 2018

			A(x) = 1 + 2*x + 20*x^2 + 352*x^3 + 8624*x^4 + 266784*x^5 + ...

Gheorghe Coserea, Table of n, a(n) for n = 0..303
Lawrence Dresner, Protection of a test magnet wound with a Ag/BSCCO high-temperature superconductor, Oak Ridge National Lab technical report (ORNL/HTSPC-3), 1992. See Eq. (25).
Noam Zeilberger, Linear lambda terms as invariants of rooted trivalent maps, arXiv:1512.06751 [cs.LO], 2015.
Noam Zeilberger, A theory of linear typings as flows on 3-valent graphs, arXiv:1804.10540 [cs.LO], 2018.
Noam Zeilberger, A Sequent Calculus for a Semi-Associative Law, arXiv preprint 1803.10030, March 2018 (A revised version of a 2017 conference paper)
Noam Zeilberger, A proof-theoretic analysis of the rotation lattice of binary trees, Part 1 (video), Rutgers Experimental Math Seminar, Sep 13 2018. Part 2 is vimeo.com/289910554.
Noam Zeilberger, From Lambda Calculus to the Four Color Theorem, via Experimental Mathematics (slides), Rutgers Experimental Math Seminar, Jun 18 2020. For the video see http://noamz.org/videos/expmath.2020.06.18.mp4.
Noam Zeilberger, From Lambda Calculus to the Four Color Theorem, via Experimental Mathematics (slides), Rutgers Experimental Math Seminar, Jun 18 2020. [Local copy]

Sequences mentioned in the Noam Zeilberger 2018 video: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827.

Cf. A000309, A062980.

a[0] = 1; a[1] = 2; a[n_] := a[n] = (6n-2) a[n-1] + Sum[(6k+2) a[k] a[n-1-k], {k, 1, n-2}];
Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Oct 16 2018, after Gheorghe Coserea *)

seq(N) = {
  my(a = vector(N)); a[1] = 2;
  for(n=2, N,
    a[n] = (6*n-2)*a[n-1] + sum(k=1, n-2, (6*k+2)*a[k]*a[n-1-k]));
  concat(1,a);
};
seq(16)
\\ test 1: y = x^2*subst(Ser(seq(201)),'x,-'x^6); 0 == x^5*y*y' + y - x^2
\\ test 2: y = Ser(seq(201)); 0 == 6*y*y'*x^2 + 2*y^2*x - y + 1
\\ Gheorghe Coserea, Nov 10 2017
F(N) = {
  my(x='x+O('x^N), t='t, F0=x, F1=0, n=1);
  while(n++,
    F1 = t + x*(F0 - subst(F0,t,0))^2 + x*deriv(F0,t);
    if (F1 == F0, break()); F0 = F1;);
  F0;
};
seq(N) = my(v=Vec(subst(F(2*N+2),'t,0))); vector((#v+1)\2, n, v[2*n-1]);
seq(16) \\ Gheorghe Coserea, Apr 01 2017

The o.g.f. f(z) = z + 2*z^3 + 20*z^5 + 352*z^7 + ... can be defined using a catalytic variable as f(z) = F(z,0), where F(z,x) satisfies the functional-differential equation F(z,x) = x + z*(F(z,x) - F(z,0))^2 + z*(d/dx)F(z,x).
From Gheorghe Coserea, Nov 10 2017: (Start)
0 = x^5*y*y' + y - x^2, where y(x) = x^2*A(-x^6).
0 = 6*y*y'*x^2 + 2*y^2*x - y + 1, where y(x) = A(x).
a(n) = (6*n-2)*a(n-1) + Sum_{k=1..n-2} (6*k+2)*a(k)*a(n-1-k), for n >= 2.
(End)
a(n) = A291843(3*n+1, 2*n), n >= 1. - Danny Rorabaugh, Nov 10 2017

cp's OEIS Frontend

A097872 Numerator of J(n) = A000356(n)/A000309(n) (average number of 4-colorings of rank 0 in a rooted nonseparable map which is trivalent and has 2n nodes).

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A097875 Denominator of J(n) = A000356(n)/A000309(n) (average number of 4-colorings of rank 0 in a rooted nonseparable map which is trivalent and has 2n nodes).

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A000699 Number of irreducible chord diagrams with 2n nodes.

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A000698 A problem of configurations: a(0) = 1; for n>0, a(n) = (2n-1)!! - Sum_{k=1..n-1} (2k-1)!! a(n-k). Also the number of shellings of an n-cube, divided by 2^n n!.

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A000260 Number of rooted simplicial 3-polytopes with n+3 nodes; or rooted 3-connected triangulations with 2n+2 faces; or rooted 3-connected trivalent maps with 2n+2 vertices.

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A000168 a(n) = 2*3^n*(2*n)!/(n!*(n+2)!).

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A000168 a(n) = 23^n(2n)!/(n!(n+2)!).

A000139 a(n) = 2(3n)! / ((2n+1)!(n+1)!).

A062980 a(0) = 1, a(1) = 5; for n > 1, a(n) = 6na(n-1) + Sum_{k=1..n-2} a(k)*a(n-k-1).