A000699 -id:A000699 - cp's OEIS frontend

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

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

A003436 Number of inequivalent labeled Hamiltonian circuits on n-octahedron. Interlacing chords joining 2n points on circle.

Original entry on oeis.org

1, 0, 1, 4, 31, 293, 3326, 44189, 673471, 11588884, 222304897, 4704612119, 108897613826, 2737023412199, 74236203425281, 2161288643251828, 67228358271588991, 2225173863019549229, 78087247031912850686, 2896042595237791161749, 113184512236563589997407

Offset: 0

N. J. A. Sloane

Also called the relaxed ménage problem (cf. A000179).
a(n) can be seen as a subset of the unordered pairings of the first 2n integers (A001147) with forbidden pairs (1,2n) and (i,i+1) for all i in [1,2n-1] (all adjacent integers modulo 2n). The linear version of this constraint is A000806. - Olivier Gérard, Feb 08 2011
Number of perfect matchings in the complement of C_{2n} where C_{2n} is the cycle graph on 2n vertices. - Andrew Howroyd, Mar 15 2016
Also the number of 2-uniform set partitions of {1...2n} containing no two cyclically successive vertices in the same block. - Gus Wiseman, Feb 27 2019

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
F. R. Bernhart & N. J. A. Sloane, Emails, April-May 1994
Kenneth P. Bogart and Peter G. Doyle, Nonsexist solution of the menage problem, Amer. Math. Monthly 93:7 (1986), 514-519.
Robert Cori and G. Hetyei, Counting partitions of a fixed genus, arXiv preprint arXiv:1710.09992 [math.CO], 2017.
M. Hazewinkel and V. V. Kalashnikov, Counting Interlacing Pairs on the Circle, CWI report AM-R9508 (1995)
Evgeniy Krasko, Igor Labutin, and Alexander Omelchenko, Enumeration of Labelled and Unlabelled Hamiltonian Cycles in Complete k-partite Graphs, arXiv:1709.03218 [math.CO], 2017.
E. Krasko and A. Omelchenko, Enumeration of Chord Diagrams without Loops and Parallel Chords, arXiv preprint arXiv:1601.05073 [math.CO], 2016.
E. Krasko and A. Omelchenko, Enumeration of Chord Diagrams without Loops and Parallel Chords, The Electronic Journal of Combinatorics, 24(3) (2017), #P3.43.
D. Singmaster, Hamiltonian circuits on the n-dimensional octahedron, J. Combinatorial Theory Ser. B 19 (1975), no. 1, 1-4.
Gus Wiseman, The a(5) = 293 interlacing chord diagrams.

Cf. A003435, A129348. A003437 gives unlabeled case.
First differences of A000806.
Column k=2 of A324428.
Cf. A000179, A000296, A000699, A001147, A005493, A170941, A190823, A278990, A306386, A306419, A322402, A324011, A324172, A324173.

A003436 := proc(n) local k;
      if n = 0 then 1
    elif n = 1 then 0
    else add( (-1)^k*binomial(n,k)*2*n/(2*n-k)*2^k*(2*n-k)!/2^n/n!,k=0..n) ;
    end if;
end proc: # R. J. Mathar, Dec 11 2013
A003436 := n-> `if`(n<2, 1-n, (-1)^n*2*hypergeom([n, -n], [], 1/2)):
seq(simplify(A003436(n)), n=0..18); # Peter Luschny, Nov 10 2016

a[n_] := (2*n-1)!! * Hypergeometric1F1[-n, 1-2*n, -2]; a[1] = 0;
Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Apr 05 2013 *)
twounifll[{}]:={{}};twounifll[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@twounifll[Complement[set,s]]]/@Table[{i,j},{j,If[i==1,Select[set,2<#i+1&]]}];
Table[Length[twounifll[Range[n]]],{n,0,14,2}] (* Gus Wiseman, Feb 27 2019 *)

a(n) = A003435(n)/(n!*2^n).
a(n) = 2*n*a(n-1)-2*(n-3)*a(n-2)-a(n-3) for n>4. [Corrected by Vasu Tewari, Apr 11 2010, and by R. J. Mathar, Oct 02 2013]
G.f.: x + ((1-x)/(1+x)) * Sum_{n>=0} A001147(n)*(x/(1+x)^2)^n. - Vladeta Jovovic, Jun 27 2007
a(n) ~ 2^(n+1/2)*n^n/exp(n+1). - Vaclav Kotesovec, Aug 13 2013
a(n) = (-1)^n*2*hypergeom([n, -n], [], 1/2) for n >= 2. - Peter Luschny, Nov 10 2016

a(0)=1 prepended by Gus Wiseman, Feb 27 2019

A278990 Number of loopless linear chord diagrams with n chords.

Original entry on oeis.org

1, 0, 1, 5, 36, 329, 3655, 47844, 721315, 12310199, 234615096, 4939227215, 113836841041, 2850860253240, 77087063678521, 2238375706930349, 69466733978519340, 2294640596998068569, 80381887628910919255, 2976424482866702081004, 116160936719430292078411

Offset: 0

N. J. A. Sloane, Dec 07 2016

nonn

See the signed version of these numbers, A000806, for much more information about these numbers.
From Gus Wiseman, Feb 27 2019: (Start)
Also the number of 2-uniform set partitions of {1..2n} containing no two successive vertices in the same block. For example, the a(3) = 5 set partitions are:
  {{1,3},{2,5},{4,6}}
  {{1,4},{2,5},{3,6}}
  {{1,4},{2,6},{3,5}}
  {{1,5},{2,4},{3,6}}
  {{1,6},{2,4},{3,5}}
(End)
From Gus Wiseman, Jul 05 2020: (Start)
Also the number of permutations of the multiset {1,1,2,2,...,n,n} with no two consecutive terms equal and where the first i appears before the first j for i < j. For example, the a(3) = 5 permutations are the following.
  (1,2,3,1,2,3)
  (1,2,3,1,3,2)
  (1,2,3,2,1,3)
  (1,2,3,2,3,1)
  (1,2,1,3,2,3)
(End)

Seiichi Manyama, Table of n, a(n) for n = 0..404 (terms 0..200 from Gheorghe Coserea)
Dmitry Efimov, The hafnian of Toeplitz matrices of a special type, perfect matchings and Bessel polynomials, arXiv:1904.08651 [math.CO], 2019.
H. Eriksson and A. Martin, Enumeration of Carlitz multipermutations, arXiv:1702.04177 [math.CO], 2017.
E. Krasko, I. Labutin, and A. Omelchenko, Enumeration of labelled and unlabelled Hamiltonian Cycles in complete k-partite graphs, arXiv:1709.03218 [math.CO], 2017, Table 1.
E. Krasko and A. Omelchenko, Enumeration of Chord Diagrams without Loops and Parallel Chords, arXiv:1601.05073 [math.CO], 2016.
E. Krasko and A. Omelchenko, Enumeration of Chord Diagrams without Loops and Parallel Chords, The Electronic Journal of Combinatorics, 24(3) (2017), #P3.43.
Gus Wiseman, The a(4) = 36 loopless linear chord diagrams.
Donovan Young, Counting Bubbles in Linear Chord Diagrams, arXiv:2311.01569 [math.CO], 2023.
Donovan Young, Bubbles in Linear Chord Diagrams: Bridges and Crystallized Diagrams, arXiv:2408.17232 [math.CO], 2024.

Column k=0 of A079267.
Column k=2 of A293157.
Row n=2 of A322013.
Cf. A000110, A000699 (topologically connected 2-uniform), A000806, A001147 (2-uniform), A003436 (cyclical version), A005493, A170941, A190823 (distance 3+ version), A322402, A324011, A324172.
Anti-run compositions are A003242.
Separable partitions are A325534.
Cf. A007716, A292884, A333489.
Other sequences involving the multiset {1,1,2,2,...,n,n}: A001147, A007717, A020555, A094574, A316972.

[n le 2 select 2-n else (2*n-3)*Self(n-1) + Self(n-2): n in [1..30]]; // G. C. Greubel, Sep 26 2023

RecurrenceTable[{a[n]== (2n-1)a[n-1] +a[n-2], a[0]==1, a[1]==0}, a, {n,0,20}] (* Vaclav Kotesovec, Sep 15 2017 *)
FullSimplify[Table[-I*(BesselI[1/2+n,-1] BesselK[3/2,1] - BesselI[3/2,-1] BesselK[1/2+ n,1]), {n,0,20}]] (* Vaclav Kotesovec, Sep 15 2017 *)
Table[(2 n-1)!! Hypergeometric1F1[-n,-2 n,-2], {n,0,20}] (* Eric W. Weisstein, Nov 14 2018 *)
Table[Sqrt[2/Pi]/E ((-1)^n Pi BesselI[1/2+n,1] +BesselK[1/2+n,1]), {n,0,20}] // FunctionExpand // FullSimplify (* Eric W. Weisstein, Nov 14 2018 *)
twouniflin[{}]:={{}};twouniflin[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@twouniflin[Complement[set,s]]]/@Table[{i,j},{j,Select[set,#>i+1&]}];
Table[Length[twouniflin[Range[n]]],{n,0,14,2}] (* Gus Wiseman, Feb 27 2019 *)

seq(N) = {
  my(a = vector(N)); a[1] = 0; a[2] = 1;
  for (n = 3, N, a[n] = (2*n-1)*a[n-1] + a[n-2]);
  concat(1, a);
};
seq(20) \\ Gheorghe Coserea, Dec 09 2016

def A278990_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(-1+sqrt(1-2*x))/sqrt(1-2*x) ).egf_to_ogf().list()
A278990_list(30) # G. C. Greubel, Sep 26 2023

From Gheorghe Coserea, Dec 09 2016: (Start)
D-finite with recurrence a(n) = (2*n-1)*a(n-1) + a(n-2), with a(0) = 1, a(1) = 0.
E.g.f. y satisfies: 0 = (1-2*x)*y'' - 3*y' - y.
a(n) - a(n-1) = A003436(n) for all n >= 2. (End)
From Vaclav Kotesovec, Sep 15 2017: (Start)
a(n) = sqrt(2)*exp(-1)*(BesselK(1/2 + n, 1)/sqrt(Pi) - i*sqrt(Pi)*BesselI(1/2 + n, -1)), where i is the imaginary unit.
a(n) ~ 2^(n+1/2) * n^n  / exp(n+1). (End)
a(n) = A114938(n)/n! - Gus Wiseman, Jul 05 2020 (from Alexander Burstein's formula at A114938).
From G. C. Greubel, Sep 26 2023: (Start)
a(n) = (-1)^n * (i/e)*Sqrt(2/Pi) * BesselK(n + 1/2, -1).
G.f.: sqrt(Pi/(2*x)) * exp(-(1+x)^2/(2*x)) * Erfi((1+x)/sqrt(2*x)).
E.g.f.: exp(-1 + sqrt(1-2*x))/sqrt(1-2*x). (End)

a(0)=1 prepended by Gheorghe Coserea, Dec 09 2016

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

A136653 G.f.: A(x) satisfies: coefficient of x^n in A(x)^(n+1)/(n+1) = 2^(n*(n-1)/2).

Original entry on oeis.org

1, 1, 1, 4, 39, 748, 27162, 1880872, 252273611, 66358216668, 34506398937158, 35644762692112792, 73356520492898454022, 301274559225693420690360, 2471654510727312089903896948, 40527708183358718551543295827536, 1328579216048284168977214446788083699

Offset: 0

Paul D. Hanna, Jan 15 2008

nonn

a(n) is the number of graphs on vertices 1,...,n such that, when these vertices are arranged counterclockwise around a circle and edges are drawn as straight line segments, the resulting diagram is connected. - Jonathan Novak (j2novak(AT)math.uwaterloo.ca), Apr 30 2010

In this interpretation, both intersecting (set theoretically) and crossing (topologically) edges are considered connected. - Gus Wiseman, Feb 23 2019

			G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 39*x^4 + 748*x^5 + 27162*x^6 +...
Let F(x) = 1 + x + 2*x^2 + 8*x^3 + 64*x^4 + 1024*x^5 +...+ 2^(n*(n-1)/2)*x^n +..
then A(x) = F(x/A(x)), A(x*F(x)) = F(x).
Coefficient of x^n in A(x)^(n+1)/(n+1) = 2^(n*(n-1)/2),
as can be seen by the main diagonal in the array of
coefficients in the initial powers of A(x):
A^1: [(1), 1, 1, 4, 39, 748, 27162, 1880872, 252273611,...;
A^2: [1, (2), 3, 10, 87, 1582, 55914, 3817876, 508370795,...;
A^3: [1, 3, (6), 19, 147, 2517, 86398, 5813550, 768378627,...;
A^4: [1, 4, 10, (32), 223, 3572, 118778, 7870640, 1032387787,...;
A^5: [1, 5, 15, 50, (320), 4771, 153245, 9992130, 1300492845,...;
A^6: [1, 6, 21, 74, 444, (6144), 190023, 12181278, 1572792585,...;
A^7: [1, 7, 28, 105, 602, 7728, (229376), 14441659, 1849390375,...;
A^8: [1, 8, 36, 144, 802, 9568, 271616, (16777216), 2130394591,...;
A^9: [1, 9, 45, 192, 1053, 11718, 317112, 19192320, (2415919104),...;
dividing each diagonal term in row n by (n+1) gives 2^(n*(n-1)/2).
The diagonal above the main diagonal gives coefficients of l.g.f.:
log(F(x)) = x + 3*x^2/2 + 19*x^3/3 + 223*x^4/4 + 4771*x^5/5 +...

Andrew Howroyd, Table of n, a(n) for n = 0..50
Gus Wiseman, The a(4) = 39 graphs connected by overlaps or crossings.

Cf. A136652 (log(A(x))), A136654.
Cf. A000699, A002662, A006125, A007297, A016098, A054726, A099947, A306438.
Cf. A324166, A324169, A324172, A324173, A324327, A324328.

max = 15; s = x*Sum[2^(k*(k-1)/2)*x^k, {k, 0, max}] + O[x]^(max+2); x/InverseSeries[s] + O[x]^(max+1) // CoefficientList[#, x]& (* Jean-François Alcover, Sep 03 2017 *)
croXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
bicmpts[stn_]:=csm[Union[Subsets[stn,{1}],Select[Subsets[stn,{2}],Intersection@@#!={}&],Select[Subsets[stn,{2}],croXQ]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],And[Union@@#==Range[n],Length[bicmpts[#]]<=1]&]],{n,0,5}] (* Gus Wiseman, Feb 23 2019 *)

a(n)=polcoeff(x/serreverse(x*sum(k=0,n,2^(k*(k-1)/2)*x^k +x*O(x^n))),n)

G.f.: A(x) = x/Series_Reversion( x*Sum_{k=0..n} 2^(k(k-1)/2)*x^k ).

Equals the free cumulant sequence corresponding to A006125. - Jonathan Novak (j2novak(AT)math.uwaterloo.ca), Apr 30 2010

Name changed and part of prior name moved to formula section by Paul D. Hanna, Sep 19 2013

A000309 Number of rooted planar bridgeless cubic maps with 2n nodes.

Original entry on oeis.org

1, 1, 4, 24, 176, 1456, 13056, 124032, 1230592, 12629760, 133186560, 1436098560, 15774990336, 176028860416, 1990947110912, 22783499599872, 263411369705472, 3073132646563840, 36143187370967040, 428157758086840320, 5105072641718353920, 61228492804372561920

Offset: 0

N. J. A. Sloane and Robert G. Wilson v

Also counts rooted planar non-separable triangulations with 3n edges. - Valery A. Liskovets, Dec 01 2003
Equivalently, rooted planar loopless triangulations with 2n triangles. - Noam Zeilberger, Oct 06 2016
Description trees of type (2,2) with n edges. (A description tree of type (a,b) is a rooted plane tree where every internal node is labeled by an integer between a and [b + sum of labels of its children], every leaf is labeled a, and the root is labeled [b + sum of labels of its children]. See Definition 1 and Section 5.2 of Cori and Schaeffer 2003.) - Noam Zeilberger, Oct 08 2017
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

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.
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).

T. D. Noe, Table of n, a(n) for n = 0..100
Marie Albenque, Dominique Poulalhon, A Generic Method for Bijections between Blossoming Trees and Planar Maps, Electron. J. Combin., 22 (2015), #P2.38.
Dario Benedetti, Sylvain Carrozza, Reiko Toriumi, Guillaume Valette, Multiple scaling limits of U(N)^2 X O(D) multi-matrix models, arXiv:2003.02100 [math-ph], 2020.
Olivier Bernardi, Bijective counting of Kreweras walks and loopless triangulations, Journal of Combinatorial Theory, Series A 114:5 (2007), 931-956.
Junliang Cai, Yanpei Liu, The enumeration of rooted nonseparable nearly cubic maps, Discrete Math. 207 (1999), no. 1-3, 9--24. MR1710479 (2000g:05074). See (31).
Robert Cori and Gilles Schaeffer, Description trees and Tutte formulas, Theoretical Computer Science 292:1 (2003), 165-183.
S. Dulucq and O. Guibert, Stack words, standard tableaux and Baxter permutations, Disc. Math. 157 (1996), 91-106.
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)
Hsien-Kuei Hwang, Mihyun Kang, 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.
R. C. Mullin, On counting rooted triangular maps, Canad. J. Math., v.17 (1965), 373-382.
Elena Patyukova, Taylor Rottreau, Robert Evans, Paul D. Topham, Martin J. Greenall, Hydrogen Bonding Aggregation in Acrylamide: Theory and Experiment, Macromolecules (2018) Vol. 51, No. 18, 7032-7043. Also arXiv:1805.09878 [math.CA], 2018.
W. T. Tutte, A census of Hamiltonian polygons, Canad. J. Math., 14 (1962), 402-417.
W. T. Tutte, On the enumeration of four-colored maps, SIAM J. Appl. Math., 17 (1969), 454-460.
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], 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.
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.

Cf. A000139, A000264, A000356, A002005, A006335, A358091.

List([0..20], n -> 2^(n+1)*Factorial(3*n)/(Factorial(n)* Factorial(2*n+2))); # G. C. Greubel, Nov 29 2018

[2^(n+1)*Factorial(3*n)/(Factorial(n)*Factorial(2*n+2)): n in [0..20]]; // Vincenzo Librandi, Aug 10 2014

a := n -> 2^(n+1)*(3*n)!/(n!*(2*n+2)!);
A000309 := n -> -(-2)^(n-1)*(3*n+2)*hypergeom([-3*(n+1),-n,-n+1/3], [-n-1,-n-2/3], 1): seq(simplify(A000309(n)), n = 0..21); # Peter Luschny, Oct 28 2022

f[n_] := 2^n(3n)!/((n + 1)!(2n + 1)!); Table[f[n], {n, 0, 19}] (* Robert G. Wilson v, Sep 21 2004 *)
Join[{1},RecurrenceTable[{a[1]==1,a[n]==4a[n-1] Binomial[3n,3]/ Binomial[2n+2,3]}, a[n],{n,20}]] (* Harvey P. Dale, May 11 2011 *)

a(n) = 2^(n+1)*(3*n)!/(n!*(2*n+2)!); \\ Michel Marcus, Aug 09 2014

[2^n*factorial(3*n)/(factorial(n+1)*factorial(2*n+1))for n in range(20)] # G. C. Greubel Nov 29 2018

a(n) = 2^(n-1) * A000139(n) for n > 0.
a(n) = 4*a(n-1)*binomial(3*n, 3) / binomial(2*n+2, 3).
a(n) = 2^n*(3*n)!/ ( (n+1)!*(2*n+1)! ).
G.f.: (1/(6*x)) * (hypergeom([ -2/3, -1/3],[1/2],(27/2)*x)-1). - Mark van Hoeij, Nov 02 2009
a(n) ~ 3^(3*n+1/2)/(sqrt(Pi)*2^(n+2)*n^(5/2)). - Ilya Gutkovskiy, Oct 06 2016
D-finite with recurrence (n+1)*(2*n+1)*a(n) -3*(3*n-1)*(3*n-2)*a(n-1)=0. - R. J. Mathar, Nov 02 2018
a(n) = -(-2)^(n-1)*(3*n+2)*hypergeom([-3*(n+1),-n,-n+1/3], [-n-1,-n-2/3], 1). The a(n) are values of the polynomials A358091. - Peter Luschny, Oct 28 2022
From Karol A. Penson, Feb 24 2025: (Start)
G.f.: hypergeom([1/3, 2/3, 1], [3/2, 2], (27*z)/2).
G.f. A(z) satisfies: - 1 + 27*z + (-36*z + 1)*A(z) + 8*z*A(z)^2 + 16*z^2*A(z)^3 = 0.
G.f.: ((4*sqrt(4 - 54*z) + 12*i*sqrt(6)*sqrt(z))^(1/3)*(sqrt(z*(4 - 54*z)) - 9*i*sqrt(6)*z) + (4*sqrt(4 - 54*z) - 12*i*sqrt(6)*sqrt(z))^(1/3)*(9*i*sqrt(6)*z + sqrt(z*(4 - 54*z))) - 8*sqrt(z))/(48*z^(3/2)), where i = sqrt(-1) is the imaginary unit.
a(n) = Integral_{x=0..27/2} x^n*W(x), where W(x) = (6^(1/3)*(9 + sqrt(81 - 6*x))^(2/3)*(9*sqrt(3) - sqrt(27 - 2*x)) - 2^(2/3)*3^(1/6)*(27 + sqrt(81 - 6*x))*x^(1/3))/(48*Pi*(9 + sqrt(81 - 6*x))^(1/3)*x^(2/3)).
This integral representation is unique as W(x) is the solution of the Hausdorff power moment problem for x on (0, 27/2). Using only the definition of a(n), W(x) can be proven to be positive. W(x) is singular at x = 0, with singularity x^(-2/3), and for x > 0 is monotonically decreasing to zero at x = 27/2. (End)

Definition clarified by Michael Albert, Oct 24 2008

A326293 Number of non-nesting, topologically connected simple graphs with vertices {1..n}.

Original entry on oeis.org

1, 1, 2, 4, 8, 27, 192, 1750

Offset: 0

Gus Wiseman, Jun 29 2019

Two edges {a,b}, {c,d} are crossing if a < c < b < d or c < a < d < b, and nesting if a < c < d < b or c < a < b < d. A graph with positive integer vertices is topologically connected if the graph whose vertices are the edges and whose edges are crossing pairs of edges is connected.

Gus Wiseman, The a(5) = 27 non-nesting, topologically connected simple graphs.

The inverse binomial transform is the covering case A326349.
Topologically connected simple graphs are A324328.
Non-crossing simple graphs are A054726.
Topologically connected set partitions are A099947.
Cf. A000108, A000699, A006125, A007297, A117662, A136653.
Cf. A324173, A324323, A324327, A326244, A326330, A326341.

croXQ[eds_]:=MatchQ[eds,{_,{x_,y_},_,{z_,t_},_}/;x_,{x_,y_},_,{z_,t_},_}/;x0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],!nesXQ[#]&&Length[csm[Union[Subsets[#,{1}],Select[Subsets[#,{2}],croXQ]]]]<=1&]],{n,0,5}]

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

A324327 Number of topologically connected chord graphs covering {1,...,n}.

Original entry on oeis.org

1, 0, 1, 0, 1, 11, 257

Offset: 0

Gus Wiseman, Feb 22 2019

A graph is topologically connected if the graph whose vertices are the edges and whose edges are crossing pairs of edges is connected, where two edges cross each other if they are of the form {{x,y},{z,t}} with x < z < y < t or z < x < t < y.

Covering means there are no isolated vertices.

			The a(0) = 1 through a(5) = 11 graphs:
  {}  {{12}}  {{13}{24}}  {{13}{14}{25}}
                          {{13}{24}{25}}
                          {{13}{24}{35}}
                          {{14}{24}{35}}
                          {{14}{25}{35}}
                          {{13}{14}{24}{25}}
                          {{13}{14}{24}{35}}
                          {{13}{14}{25}{35}}
                          {{13}{24}{25}{35}}
                          {{14}{24}{25}{35}}
                          {{13}{14}{24}{25}{35}}

Gus Wiseman, The a(5) = 11 topologically connected chord graphs.
Gus Wiseman, The a(6) = 257 topologically connected chord graphs.

Cf. A000108, A000699 (the case with disjoint edges), A001764, A002061, A007297, A016098, A054726, A099947, A136653 (the case with set-theoretical connectedness also), A268814.

Cf. A324167, A324169 (non-crossing covers), A324172, A324173, A324323, A324328 (non-covering case).

croXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
crosscmpts[stn_]:=csm[Union[Subsets[stn,{1}],Select[Subsets[stn,{2}],croXQ]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],And[Union@@#==Range[n],Length[crosscmpts[#]]<=1]&]],{n,0,5}]

Inverse binomial transform of A324328.

A002005 Number of rooted planar cubic maps with 2n vertices.

Original entry on oeis.org

1, 4, 32, 336, 4096, 54912, 786432, 11824384, 184549376, 2966845440, 48855252992, 820675092480, 14018773254144, 242919827374080, 4261707069259776, 75576645116559360, 1353050213048123392, 24428493151359467520, 444370175232646840320, 8138178004138611179520

Offset: 0

N. J. A. Sloane

nonn

Equivalently, number of rooted planar triangulations with 2n faces.

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

R. C. Mullin, E. Nemeth and P. J. Schellenberg, The enumeration of almost cubic maps, pp. 281-295 in Proceedings of the Louisiana Conference on Combinatorics, Graph Theory and Computer Science. Vol. 1, edited R. C. Mullin et al., 1970.
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).

Gheorghe Coserea, Table of n, a(n) for n = 0..1000
Valentin Bonzom, Guillaume Chapuy, Maciej Dolega, Enumeration of non-oriented maps via integrability, Alg. Combin. 5 (6) (2022) p 1363-1390, A.3
Mireille Bousquet-Mélou, Counting planar maps, coloured or uncoloured, 23rd British Combinatorial Conference, Jul 2011, Exeter, United Kingdom. 392, pp.1-50, 2011, London Math. Soc. Lecture Note Ser., hal-00653963. See p.13.
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].
Maxim Krikun, Explicit enumeration of triangulations with multiple boundaries, arXiv:0706.0681 [math.CO], 2007. [Comment from _Gheorghe Coserea_, Dec 26 2015: the formula in the paper for almost trivalent maps is 2 * 4^(k-1) * (3k)!!/ ((k+1)!*(k+2)!!); however, the exponent of 4 should be k not (k-1) i.e. 2 * 4^k * (3k)!! / ((k+1)!*(k+2)!!)]
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.
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.
Column k=0 of A266240.
Cf. A000217, A358367.

seq(2*8^n*binomial(n*3/2, n)/((n + 2)*(n + 1)), n = 0..19); # Peter Luschny, Nov 14 2022

Table[2^(2 n + 1) (3 n)!!/((n + 2)! n!!), {n, 0, 20}] (* Vincenzo Librandi, Dec 28 2015 *)
CoefficientList[Series[(-1 + 96 z + Hypergeometric2F1[-2/3,-1/3,1/2,432z^2]- 96 z Hypergeometric2F1[-1/6,1/6,3/2,432z^2])/(192 z^2), {z, 0, 10}], z] (* Benedict W. J. Irwin, Aug 07 2016 *)

factorial2(n) = my(x = (2^(n\2)*(n\2)!)); if (n%2, n!/x, x);
a(n) = 2^(2*n+1)*factorial2(3*n)/((n+2)!*factorial2(n));
vector(20, i, a(i-1))
\\ test: y = Ser(vector(201, n, a(n-1))); x*(1-432*x^2)*y' == 64*x^2*y^2 + (288*x^2 - 64*x - 1)*y + 72*x + 1
\\ Gheorghe Coserea, Jun 13 2017

a(n) = 2^(2*n+1)*(3*n)!!/((n+2)!*n!!). - Sean A. Irvine, May 19 2013
a(n) ~ sqrt(6/Pi) * n^(-5/2) * (12*sqrt(3))^n. - Gheorghe Coserea, Feb 25 2016
G.f.: (96*x - 1 + 2F1(-2/3, -1/3; 1/2; 432*x^2) - 96*x*2F1(-1/6, 1/6; 3/2; 432*x^2))/(192*x^2). - Benedict W. J. Irwin, Aug 07 2016
From Gheorghe Coserea, Jun 13 2017: (Start)
G.f. y(x) satisfies:
x*(1-432*x^2)*deriv(y,x) = 64*x^2*y^2 + (288*x^2 - 64*x - 1)*y + 72*x + 1.
0 = 64*x^3*y^3 + x*(1-96*x)*y^2 + (30*x-1)*y - 27*x + 1.
(End).
D-finite with recurrence (n+2)*(n+1)*a(n) -48*(3*n-2)*(3*n-4)*a(n-2)=0. - R. J. Mathar, Feb 08 2021
From Karol A. Penson and Katarzyna Gorska (katarzyna.gorska@ifj.edu.pl), Nov 02 2022: (Start)
a(n) = Integral_{x=0..12*sqrt(3)} x^n*W(x), where
W(x) = (T1(x) + T2(x)) / T3(x), and
T1(x) = -x^(2/3) * (108 + sqrt(3) * sqrt(432 - x^2));
T2(x) = 3^(1/6)*(36+sqrt(3)*sqrt(432-x^2))^(2/3) * (-432+x^2+36*sqrt(3)* sqrt(432-x^2)) / sqrt(432-x^2);
T3(x) = (128*3^(5/6)*Pi*x^(1/3)*(36+sqrt(3)*sqrt(432-x^2))^(1/3)).
This integral representation is unique as W(x) is the solution of the Hausdorff power moment problem. Using only the definition of a(n), W(x) can be proven to be positive. W(x) is singular at x = 0, with the singularity x^(-1/3), and for x > 0 is monotonically decreasing to zero at x = 12*sqrt(3). (End)
a(n) = 2^(3*n + 1)*binomial(n*3/2, n)/((n + 1)*(n + 2)) = A358367(n) / A000217(n + 1). - Peter Luschny, Nov 14 2022

More terms from Sean A. Irvine, May 19 2013

cp's OEIS Frontend

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

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A003436 Number of inequivalent labeled Hamiltonian circuits on n-octahedron. Interlacing chords joining 2n points on circle.

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A278990 Number of loopless linear chord diagrams with n chords.

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

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A136653 G.f.: A(x) satisfies: coefficient of x^n in A(x)^(n+1)/(n+1) = 2^(n*(n-1)/2).

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A000309 Number of rooted planar bridgeless cubic maps with 2n nodes.

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

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).