A000260 -id:A000260 - cp's OEIS frontend

A002293 Number of dissections of a polygon: binomial(4n, n)/(3n + 1).

1, 1, 4, 22, 140, 969, 7084, 53820, 420732, 3362260, 27343888, 225568798, 1882933364, 15875338990, 134993766600, 1156393243320, 9969937491420, 86445222719724, 753310723010608, 6594154339031800, 57956002331347120, 511238042454541545

Offset: 0

N. J. A. Sloane

The number of rooted loopless n-edge maps in the plane (planar with a distinguished outside face). - Valery A. Liskovets, Mar 17 2005
Number of lattice paths from (1,0) to (3*n+1,n) which, starting from (1,0), only utilize the steps +(1,0) and +(0,1) and additionally, the paths lie completely below the line y = (1/3)*x (i.e., if (a,b) is in the path, then b < a/3). - Joseph Cooper (jecooper(AT)mit.edu), Feb 07 2006
Number of length-n restricted growth strings (RGS) [s(0), s(1), ..., s(n-1)] where s(0) = 0 and s(k) <= s(k-1) + 3, see fxtbook link below. - Joerg Arndt, Apr 08 2011
From Wolfdieter Lang, Sep 14 2007: (Start)
a(n), n >= 1, enumerates quartic trees (rooted, ordered, incomplete) with n vertices (including the root).
Pfaff-Fuss-Catalan sequence C^{m}_n for m = 4. See the Graham et al. reference, p. 347. eq. 7.66. (Second edition, p. 361, eq. 7.67.) See also the Pólya-Szegő reference.
Also 4-Raney sequence. See the Graham et al. reference, pp. 346-347.
(End)
Bacher: "We describe the statistics of checkerboard triangulations obtained by coloring black every other triangle in triangulations of convex polygons." The current sequence (A002293) occurs on p. 12 as one of two "extremal sequences" of an array of coefficients of polynomials, whose generating functions are given in terms of hypergeometric functions. - Jonathan Vos Post, Oct 05 2007
A generating function in terms of a (labyrinthine) solution to a depressed quartic equation is given in the Copeland link for signed A005810. With D(z,t) that g.f., a g.f. for signed A002293 is {[-1+1/D(z,t)]/(4t)}^(1/3). - Tom Copeland, Oct 10 2012
For a relation to the inviscid Burgers's equation, see A001764. - Tom Copeland, Feb 15 2014
For relations to compositional inversion, the Legendre transform, and convex geometry, see the Copeland, the Schuetz and Whieldon, and the Gross (p. 58) links. - Tom Copeland, Feb 21 2017 (See also Gross et al. in A062994. - Tom Copeland, Dec 24 2019)
This is the number of A'Campo bicolored forests of degree n and co-dimension 0. This can be shown using generating functions or a combinatorial approach. See Combe and Jugé link below. - Noemie Combe, Feb 28 2017
Conjecturally, a(n) is the number of 3-uniform words over the alphabet [n] that avoid the patterns 231 and 221 (see the Defant and Kravitz link). - Colin Defant, Sep 26 2018
The compositional inverse o.g.f. pair in Copeland's comment above are related to a pair of quantum fields in Balduf's thesis by Theorem 4.2 on p. 92. Cf. A001764. - Tom Copeland, Dec 13 2019
a(n) is the total number of down steps before the first up step in all 3_1-Dyck paths of length 4*n. A 3_1-Dyck path is a lattice path with steps (1, 3), (1, -1) that starts and ends at y = 0 and stays above the line y = -1. - Sarah Selkirk, May 10 2020
a(n) is the number of pairs (A<=B) of noncrossing partitions of [2n] such that every block of A has exactly two elements. In fact, it is proved that a(n) is the number of planar tied arc diagrams with n arcs (see Aicardi link below). A planar diagram with n arcs represents a noncrossing partition A of [2n] with n blocks, each block containing the endpoints of one arc; each tie connects two arcs, so that the ties define a partition B >= A: the endpoints of two arcs connected by a tie belong to the same block of B. Ties do not cross arcs nor other ties iff B has a planar diagram, i.e., B is a noncrossing partition. - Francesca Aicardi, Nov 07 2022
Dropping the initial 1 (starting 1, 4, 22 with offset 1) yields the REVERT transformation 1, -4 ,10, -20, 35.. essentially A000292 without leading 0. - R. J. Mathar, Aug 17 2023
Number of rooted polyominoes composed of n pentagonal cells of the hyperbolic regular tiling with Schläfli symbol {5,oo}. A rooted polyomino has one external edge identified, and chiral pairs are counted as two. A stereographic projection of the {5,oo} tiling on the Poincaré disk can be obtained via the Christensson link. - Robert A. Russell, Jan 27 2024
This is instance k = 4 of the generalized Catalan family {C(k, n)}A130564.%20-%20_Wolfdieter%20Lang">{n>=0} given in a comment of A130564. - _Wolfdieter Lang, Feb 05 2024
a(n) is the cardinality of the planar ramified Jones monoid PR(J_n). - Diego Arcis, Nov 21 2024

			There are a(2) = 4 quartic trees (vertex degree <= 4 and 4 possible branchings) with 2 vertices (one of them the root). Adding one more branch (one more vertex) to these four trees yields 4*4 + 6 = 22 = a(3) such trees.

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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..1000[Terms 0 to 100 computed by T. D. Noe; terms 101 to 1000 by G. C. Greubel, Jan 14 2017]
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Francesca Aicardi, Catalan triangle and tied arc diagrams, arXiv:2011.14628 [math.CO], 2020.
Francesca Aicardi, Diego Arcis, and Jesús Juyumaya, Ramified inverse and planar monoids, Mosc Math J, 24(3):321-355, 9 2024.
M. Almeida, N. Moreira, and R. Reis, Enumeration and generation with a string automata representation, Theor. Comp. Sci. 387 (2007) 93-102, Theor. 6
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A. Asinowski, B. Hackl, and S. Selkirk, Down step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.
Roland Bacher, Fair Triangulations, arXiv:0710.0960 [math.CO], 2007.
P. Balduf, The propagator and diffeomorphisms of an interacting field theory, Master's thesis, submitted to the Institut für Physik, Mathematisch-Naturwissenschaftliche Fakultät, Humboldt-Universtät, Berlin, 2018.
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Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.
Paul Barry, On the Gap-sum and Gap-product Sequences of Integer Sequences, arXiv:2104.05593 [math.CO], 2021.
Paul Barry, d-orthogonal polynomials, Fuss-Catalan matrices and lattice paths, arXiv:2505.16718 [math.CO], 2025. See p. 2.
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D. Bevan, D. Levin, P. Nugent, J. Pantone, and L. Pudwell, Pattern avoidance in forests of binary shrubs, arXiv preprint arXiv:1510:08036 [math.CO], 2015.
Michel Bousquet and Cédric Lamathe, On symmetric structures of order two, Discrete Math. Theor. Comput. Sci. 10 (2008), 153-176.
T. Daniel Brennan, Christian Ferko, and Savdeep Sethi, A Non-Abelian Analogue of DBI from T₸, arXiv:1912.12389 [hep-th], 2019. See also SciPost Phys. Vol. 8 (2020), 052.
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Malin Christensson, Make hyperbolic tilings of images, web page, 2019.
J. Cigler, Some remarks about q-Chebyshev polynomials and q-Catalan numbers and related results, 2013.
N. Combe and V. Jugé, Counting bi-colored A'Campo forests, arXiv:1702.07672 [math.AG], 2017.
Tom Copeland, Discriminating Deltas, Depressed Equations, and Generalized Catalan Numbers, 2012.
Olivier Danvy, Summa Summarum: Moessner's Theorem without Dynamic Programming, arXiv:2412.03127 [cs.DM], 2024. See p. 25.
C. Defant and N. Kravitz, Stack-sorting for words, arXiv:1809.09158 [math.CO], 2018.
Isaac DeJager, Madeleine Naquin, and Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
Bryan Ek, Lattice Walk Enumeration, arXiv:1803.10920 [math.CO], 2018.
Bryan Ek, Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics, arXiv:1804.05933 [math.CO], 2018.
Jishe Feng, The Hessenberg matrices and Catalan and its generalized numbers, arXiv:1810.09170 [math.CO], 2018. See p. 4.
M. Gross, Mirror symmetry and the Strominger-Yau-Zaslow conjecture, arXiv:1212.4220 [math.AG], p. 58, 2013.
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Clemens Heuberger, Sarah J. Selkirk, and Stephan Wagner, Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo k, arXiv:2204.14023 [math.CO], 2022.
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Ionut E. Iacob, T. Bruce McLean and Hua Wang, The V-flex, Triangle Orientation, and Catalan Numbers in Hexaflexagons, The College Mathematics Journal, Vol. 43, No. 1 (January 2012), pp. 6-10.
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Column k=3 of triangle A062993 and A070914.
Cf. A000260, A002295, A002296, A027836, A062994, A346646 (binomial transform), A346664 (inverse binomial transform).
Cf. A006632, A006633, A006634, A025174, A069271, A196678, A224274, A233658, A233666, A233667, A277877, A283049, A283101, A283102, A283103.
Polyominoes: A005038 (oriented), A005040 (unoriented), A369471 (chiral), A369472 (achiral), A001764 {4,oo}, A002294 {6,oo}.
Cf. A130564 (for generalized Catalan C(k, n), for = 4).

List([0..22],n->Binomial(4*n,n)/(3*n+1)); # Muniru A Asiru, Nov 01 2018

[ Binomial(4*n,n)/(3*n+1): n in [0..50]]; // Vincenzo Librandi, Apr 19 2011

series(RootOf(g = 1+x*g^4, g),x=0,20); # Mark van Hoeij, Nov 10 2011
seq(binomial(4*n, n)/(3*n+1),n=0..20); # Robert FERREOL, Apr 02 2015
# Using the integral representation above:
Digits:=6;
R:=proc(x)((I + sqrt(3))*(4*sqrt(256 - 27*x) - 12*I*sqrt(3)*sqrt(x))^(1/3))/16 - ((I - sqrt(3))*(4*sqrt(256 - 27*x) + 12*I*sqrt(3)*sqrt(x))^(1/3))/16;end;
W:=proc(x) x^(-3/4) * sqrt(4*R(x) - 3^(3/4)*x^(1/4)/sqrt(R(x)))/(2*3^(1/4)*Pi);end;
# Attention: W(x) is singular at x = 0. Integration is done from  a very small positive x to x = 256/27.
# For a(8):  ... gives 420732
evalf(int(x^8*W(x),x=0.000001..256/27));
# Karol A. Penson, Jul 05 2024

CoefficientList[InverseSeries[ Series[ y - y^4, {y, 0, 60}], x], x][[Range[2, 60, 3]]]
Table[Binomial[4n,n]/(3n+1),{n,0,25}] (* Harvey P. Dale, Apr 18 2011 *)
CoefficientList[1 + InverseSeries[Series[x/(1 + x)^4, {x, 0, 60}]], x] (* Gheorghe Coserea, Aug 12 2015 *)
terms = 22; A[] = 0; Do[A[x] = 1 + x*A[x]^4 + O[x]^terms, terms];
CoefficientList[A[x], x] (* Jean-François Alcover, Jan 13 2018 *)

a(n)=binomial(4*n,n)/(3*n+1) /* Charles R Greathouse IV, Jun 16 2011 */

my(x='x+O('x^33)); Vec(1 + serreverse(x/(1+x)^4)) \\ Gheorghe Coserea, Aug 12 2015

A002293_list, x = [1], 1
for n in range(100):
    x = x*4*(4*n+3)*(4*n+2)*(4*n+1)//((3*n+2)*(3*n+3)*(3*n+4))
    A002293_list.append(x) # Chai Wah Wu, Feb 19 2016

O.g.f. satisfies: A(x) = 1 + x*A(x)^4 = 1/(1 - x*A(x)^3).
a(n) = binomial(4*n,n-1)/n, n >= 1, a(0) = 1. From the Lagrange series of the o.g.f. A(x) with its above given implicit equation.
From Karol A. Penson, Apr 02 2010: (Start)
Integral representation as n-th Hausdorff power moment of a positive function on the interval [0, 256/27]:
a(n) = Integral_{x=0..256/27}(x^n((3/256) * sqrt(2) * sqrt(3) * ((2/27) * 3^(3/4) * 27^(1/4) * 256^(/4) * hypergeom([-1/12, 1/4, 7/12], [1/2, 3/4], (27/256)*x)/(sqrt(Pi) * x^(3/4)) - (2/27) * sqrt(2) * sqrt(27) * sqrt(256) * hypergeom([1/6, 1/2, 5/6], [3/4, 5/4], (27/256)*x)/ (sqrt(Pi) * sqrt(x)) - (1/81) * 3^(1/4) * 27^(3/4) * 256^(1/4) * hypergeom([5/12, 3/4, 13/12], [5/4, 3/2], (27/256)*x/(sqrt(Pi)*x^(1/4)))/sqrt(Pi))).
This representation is unique as it represents the solution of the Hausdorff moment problem.
O.g.f.: hypergeom([1/4, 1/2, 3/4], [2/3, 4/3], (256/27)*x);
E.g.f.: hypergeom([1/4, 1/2, 3/4], [2/3, 1, 4/3], (256/27)*x). (End)
a(n) = upper left term in M^n, M = the production matrix:
  1, 1
  3, 3, 1
  6, 6, 3, 1
  ...
(where 1, 3, 6, 10, ...) is the triangular series. - Gary W. Adamson, Jul 08 2011
O.g.f. satisfies g = 1+x*g^4. If h is the series reversion of x*g, so h(x*g)=x, then (x-h(x))/x^2 is the o.g.f. of A006013. - Mark van Hoeij, Nov 10 2011
a(n) = binomial(4*n+1, n)/(4*n+1) = A062993(n+2,2). - Robert FERREOL, Apr 02 2015
a(n) = Sum_{i=0..n-1} Sum_{j=0..n-1-i} Sum_{k=0..n-1-i-j} a(i)*a(j)*a(k)*a(n-1-i-j-k) for n>=1; and a(0) = 1. - Robert FERREOL, Apr 02 2015
a(n) ~ 2^(8*n+1/2) / (sqrt(Pi) * n^(3/2) * 3^(3*n+3/2)). - Vaclav Kotesovec, Jun 03 2015
From Peter Bala, Oct 16 2015: (Start)
A(x)^2 is o.g.f. for A069271; A(x)^3 is o.g.f. for A006632;
A(x)^5 is o.g.f. for A196678; A(x)^6 is o.g.f. for A006633;
A(x)^7 is o.g.f. for A233658; A(x)^8 is o.g.f. for A233666;
A(x)^9 is o.g.f. for A006634; A(x)^10 is o.g.f. for A233667. (End)
D-finite with recurrence: a(n+1) = a(n)*4*(4*n + 3)*(4*n + 2)*(4*n + 1)/((3*n + 2)*(3*n + 3)*(3*n + 4)). - Chai Wah Wu, Feb 19 2016
E.g.f.: F([1/4, 1/2, 3/4], [2/3, 1, 4/3], 256*x/27), where F is the generalized hypergeometric function. - Stefano Spezia, Dec 27 2019
x*A'(x)/A(x) = (A(x) - 1)/(- 3*A(x) + 4) = x + 7*x^2 + 55*x^3 + 455*x^4 + ... is the o.g.f. of A224274. Cf. A001764 and A002294 - A002296. - Peter Bala, Feb 04 2022
a(n) = hypergeom([1 - n, -3*n], [2], 1). Row sums of A173020. - Peter Bala, Aug 31 2023
G.f.: t*exp(4*t*hypergeom([1, 1, 5/4, 3/2, 7/4], [4/3, 5/3, 2, 2], (256*t)/27))+1. - Karol A. Penson, Dec 20 2023
From Karol A. Penson, Jul 03 2024: (Start)
a(n) = Integral_{x=0..256/27} x^(n)*W(x)dx, n>=0, where W(x) = x^(-3/4) * sqrt(4*R(x) - 3^(3/4)*x^(1/4)/sqrt(R(x)))/(2*3^(1/4)*Pi), with R(x) = ((i + sqrt(3))*(4*sqrt(256 - 27*x) -12*i*sqrt(3*x))^(1/3))/16 - ((i - sqrt(3))*(4*sqrt(256 - 27*x) + 12*i* sqrt(3*x))^(1/3))/16, where i is the imaginary unit.
The elementary function W(x) is positive on the interval x = (0, 256/27) and is equal to the combination of hypergeometric functions in my formula from 2010; see above.
(Pi*W(x))^6 satisfies an algebraic equation of order 6, with integer polynomials as coefficients. (End)
G.f.: (Sum_{n >= 0} binomial(4*n+1, n)*x^n) / (Sum_{n >= 0} binomial(4*n, n)*x^n). - Peter Bala, Dec 14 2024
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^7). - Seiichi Manyama, Jun 16 2025

A006013 a(n) = binomial(3*n+1,n)/(n+1).

Original entry on oeis.org

1, 2, 7, 30, 143, 728, 3876, 21318, 120175, 690690, 4032015, 23841480, 142498692, 859515920, 5225264024, 31983672534, 196947587823, 1219199353190, 7583142491925, 47365474641870, 296983176369495, 1868545312633440, 11793499763070480

Offset: 0

N. J. A. Sloane

Enumerates pairs of ternary trees [Knuth, 2014]. - N. J. A. Sloane, Dec 09 2014
G.f. (offset 1) is series reversion of x - 2x^2 + x^3.
Hankel transform is A005156(n+1). - Paul Barry, Jan 20 2007
a(n) = number of ways to connect 2*n - 2 points labeled 1, 2, ..., 2*n-2 in a line with 0 or more noncrossing arcs above the line such that each maximal contiguous sequence of isolated points has even length. For example, with arcs separated by dashes, a(3) = 7 counts {} (no arcs), 12, 14, 23, 34, 12-34, 14-23. It does not count 13 because 2 is an isolated point. - David Callan, Sep 18 2007
In my 2003 paper I introduced L-algebras. These are K-vector spaces equipped with two binary operations > and < satisfying (x > y) < z = x > (y < z). In my arXiv paper math-ph/0709.3453 I show that the free L-algebra on one generator is related to symmetric ternary trees with odd degrees, so the dimensions of the homogeneous components are 1, 2, 7, 30, 143, .... These L-algebras are closely related to the so-called triplicial-algebras, 3 associative operations and 3 relations whose free object is related to even trees. - Philippe Leroux (ph_ler_math(AT)yahoo.com), Oct 05 2007
a(n-1) is also the number of projective dependency trees with n nodes. - Marco Kuhlmann (marco.kuhlmann(AT)lingfil.uu.se), Apr 06 2010
Number of subpartitions of [1^2, 2^2, ..., n^2]. - Franklin T. Adams-Watters, Apr 13 2011
a(n) = sum of (n+1)-th row terms of triangle A143603. - Gary W. Adamson, Jul 07 2011
Also the number of Dyck n-paths with up steps colored in two ways (N or A) and avoiding NA. The 7 Dyck 2-paths are NDND, ADND, NDAD, ADAD, NNDD, ANDD, and AADD. - David Scambler, Jun 24 2013
a(n) is also the number of permutations avoiding 213 in the classical sense which can be realized as labels on an increasing strict binary tree with 2n-1 nodes. See A245904 for more information on increasing strict binary trees. - Manda Riehl Aug 07 2014
With offset 1, a(n) is the number of ordered trees (A000108) with n non-leaf vertices and n leaf vertices such that every non-leaf vertex has a leaf child (and hence exactly one leaf child). - David Callan, Aug 20 2014
a(n) is the number of paths in the plane with unit east and north steps, never going above the line x=2y, from (0,0) to (2n+1,n). - Ira M. Gessel, Jan 04 2018
a(n) is the number of words on the alphabet [n+1] that avoid the patterns 231 and 221 and contain exactly one 1 and exactly two occurrences of every other letter. - Colin Defant, Sep 26 2018
a(n) is the number of Motzkin paths of length 3n with n of each type of step, such that (1, 1) and (1, 0) alternate (ignoring (-1, 1) steps). All paths start with a (1, 1) step. - Helmut Prodinger, Apr 08 2019
Hankel transform omitting a(0) is A051255(n+1). - Michael Somos, May 15 2022
If f(x) is the generating function for (-1)^n*a(n), a real solution of the equation y^3 - y^2 - x = 0 is given by y = 1 + x*f(x). In particular 1 + f(1) is Narayana's cow constant, A092526, aka the "supergolden" ratio. - R. James Evans, Aug 09 2023
This is instance k = 2 of the family {c(k, n+1)}A130564.%20_Wolfdieter%20Lang">{n>=0} described in A130564. _Wolfdieter Lang, Feb 04 2024
Also the number of quadrangulations of a (2n+4)-gon which do not contain any diagonals incident to a fixed vertex. - Esther Banaian, Mar 12 2025

			a(3) = 30 since the top row of Q^3 = (12, 12, 5, 1).
G.f. = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 728*x^5 + 3876*x^6 + 21318*x^7 + ... - _Michael Somos_, May 15 2022

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..1000 (terms n = 0..100 from T. D. Noe)
A. Aggarwal, Armstrong's Conjecture for (k, mk+1)-Core Partitions, arXiv:1407.5134 [math.CO], 2014.
C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps, FPSAC02, Melbourne, 2002.
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, (2016), #16.3.5.
Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
W. G. Brown, Enumeration of non-separable planar maps, Canad. J. Math., 15 (1963), 526-545.
W. G. Brown, Enumeration of non-separable planar maps. [Annotated scanned copy]
Naiomi Cameron and J. E. McLeod, Returns and Hills on Generalized Dyck Paths, Journal of Integer Sequences, 19 (2016), #16.6.1.
F. Cazals, Combinatorics of Non-Crossing Configurations, Studies in Automatic Combinatorics, Volume II (1997).
F. Chapoton, F. Hivert, and J.-C. Novelli, A set-operad of formal fractions and dendriform-like sub-operads, arXiv:1307.0092 [math.CO], 2013.
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Jins de Jong, Alexander Hock, and Raimar Wulkenhaar, Catalan tables and a recursion relation in noncommutative quantum field theory, arXiv:1904.11231 [math-ph], 2019.
C. Defant and N. Kravitz, Stack-sorting for words, arXiv:1809.09158 [math.CO], 2018.
Isaac DeJager, Madeleine Naquin, and Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
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These are the odd indices of A047749.
Cf. A121645, A115728, A143603, A236194.
Cf. A000260, A007318, A024485, A051255, A071948, A092526, A110616, A258708.
Cf. A305574 (the same with offset 1 and the initial 1 replaced with 5).
Cf. A130564 (comment on c(k, n+1)).

a006013 n = a007318 (3 * n + 1) n `div` (n + 1)
a006013' n = a258708 (2 * n + 1) n
-- Reinhard Zumkeller, Jun 22 2015

[Binomial(3*n+1,n)/(n+1): n in [0..30]]; // Vincenzo Librandi, Mar 29 2015

Binomial[3#+1,#]/(#+1)&/@Range[0,30]  (* Harvey P. Dale, Mar 16 2011 *)

A006013(n) = binomial(3*n+1,n)/(n+1) \\ M. F. Hasler, Jan 08 2024

from math import comb
def A006013(n): return comb(3*n+1,n)//(n+1) # Chai Wah Wu, Jul 30 2022

def A006013_list(n) :
    D = [0]*(n+1); D[1] = 1
    R = []; b = false; h = 1
    for i in range(2*n) :
        for k in (1..h) : D[k] += D[k-1]
        if b : R.append(D[h]); h += 1
        b = not b
    return R
A006013_list(23) # Peter Luschny, May 03 2012

G.f. is square of g.f. for ternary trees, A001764 [Knuth, 2014]. - N. J. A. Sloane, Dec 09 2014
Convolution of A001764 with itself: 2*C(3*n + 2, n)/(3*n + 2), or C(3*n + 2, n+1)/(3*n + 2).
G.f.: (4/(3*x)) * sin((1/3)*arcsin(sqrt(27*x/4)))^2.
G.f.: A(x)/x with A(x)=x/(1-A(x))^2. - Vladimir Kruchinin, Dec 26 2010
From Gary W. Adamson, Jul 14 2011: (Start)
a(n) is the top left term in M^n, where M is the infinite square production matrix:
   2, 1, 0, 0, 0, ...
   3, 2, 1, 0, 0, ...
   4, 3, 2, 1, 0, ...
   5, 4, 3, 2, 1, ...
   ... (End)
From Gary W. Adamson, Aug 11 2011: (Start)
a(n) is the sum of top row terms in Q^n, where Q is the infinite square production matrix as follows:
   1, 1, 0, 0, 0, ...
   2, 2, 1, 0, 0, ...
   3, 3, 2, 1, 0, ...
   4, 4, 3, 2, 1, ...
   ... (End)
D-finite with recurrence: 2*(n+1)*(2n+1)*a(n) = 3*(3n-1)*(3n+1)*a(n-1). - R. J. Mathar, Dec 17 2011
a(n) = 2*A236194(n)/n for n > 0. - Bruno Berselli, Jan 20 2014
a(n) = A258708(2*n+1, n). - Reinhard Zumkeller, Jun 22 2015
From Ilya Gutkovskiy, Dec 29 2016: (Start)
E.g.f.: 2F2([2/3, 4/3]; [3/2,2]; 27*x/4).
a(n) ~ 3^(3*n+3/2)/(sqrt(Pi)*4^(n+1)*n^(3/2)). (End)
a(n) = A110616(n+1, 1). - Ira M. Gessel, Jan 04 2018
0 = v0*(+98415*v2 -122472*v3 +32340*v4) +v1*(+444*v3 -2968*v4) +v2*(-60*v2 +56*v3 +64*v4) where v0=a(n)^2, v1=a(n)*a(n+1), v2=a(n+1)^2, v3=a(n+1)*a(n+2), v4=a(n+2)^2 for all n in Z if a(-1)=-2/3 and a(n)=0 for n<-1. - Michael Somos, May 15 2022
a(n) = (1/4^n) * Product_{1 <= i <= j <= 2*n} (2*i + j + 2)/(2*i + j - 1). Cf. A000260. - Peter Bala, Feb 21 2023
From Karol A. Penson, Jun 02 2023: (Start)
a(n) = Integral_{x=0..27/4} x^n*W(x) dx, where
W(x) = (((9 + sqrt(81 - 12*x))^(2/3) - (9 - sqrt(81 - 12*x))^(2/3)) * 2^(1/3) * 3^(1/6)) / (12 * Pi * x^(1/3)), for x in (0, 27/4).
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 = 27/4. At x = 27/4 the first derivative of W(x) is infinite. (End)
G.f.: hypergeometric([2/3,1,4/3], [3/2,2], (3^3/2^2)*x). See the e.g.f. above. - Wolfdieter Lang, Feb 04 2024
a(n) = A024485(n+1)/3. - Michael Somos, Oct 14 2024
G.f.: (Sum_{n >= 0} binomial(3*n+2, n)*x^n) / (Sum_{n >= 0} binomial(3*n, n)*x^n) = (B(x) - 1)/(x*B(x)), where B(x) = Sum_{n >= 0} binomial(3*n, n)/(2*n+1) * x^n is the g.f. of A001764. - Peter Bala, Dec 13 2024
The g.f. A(x) is uniquely determined by the conditions A(0) = 1 and [x^n] A(x)^(-n) = -2 for all n >= 1. Cf. A006632. - Peter Bala, Jul 24 2025

Edited by N. J. A. Sloane, Feb 21 2008

A000699 Number of irreducible chord diagrams with 2n nodes.

Original entry on oeis.org

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.
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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)
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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

A006632 a(n) = 3binomial(4n-1, n-1)/(4*n-1).

Original entry on oeis.org

1, 3, 15, 91, 612, 4389, 32890, 254475, 2017356, 16301164, 133767543, 1111731933, 9338434700, 79155435870, 676196049060, 5815796869995, 50318860986108, 437662920058980, 3824609516638444, 33563127932394060, 295655735395397520, 2613391671568320765

Offset: 1

Simon Plouffe

a(n) is the number of ordered trees (A000108) with 3n-1 edges in which every non-leaf vertex has exactly two leaf children (no restriction on non-leaf children). For example, a(2) counts the 3 trees
  \/......\/......\/
  .\|/...\|/....\|/  . - David Callan, Aug 22 2014
a(n) is the number of lattice paths from (0,0) to (3n,n) using only the steps (1,0) and (0,1) and which are strictly below the line y = x/3 except at the path's endpoints. - Lucas A. Brown, Aug 21 2020
This is instance k = 3 of the family {c(k, n)}A130564.%20-%20_Wolfdieter%20Lang">{n>=1} given in a comment in A130564. - _Wolfdieter Lang, Feb 04 2024

H. M. Finucan, Some decompositions of generalized Catalan numbers, pp. 275-293 of Combinatorial Mathematics IX. Proc. Ninth Australian Conference (Brisbane, August 1981). Ed. E. J. Billington, S. Oates-Williams and A. P. Street. Lecture Notes Math., 952. Springer-Verlag, 1982.
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 = 1..1000
O. Aichholzer, A. Asinowski and T. Miltzow, Disjoint compatibility graph of non-crossing matchings of points in convex position, arXiv preprint arXiv:1403.5546 [math.CO], 2014.
Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.
Clemens Heuberger, Sarah J. Selkirk, and Stephan Wagner, Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo k, arXiv:2204.14023 [math.CO], 2022.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 438
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
J. Sawada, J. Sears, A. Trautrim, and A. Williams, Demystifying our Grandparent's De Bruijn Sequences with Concatenation Trees, arXiv:2308.12405 [math.CO], 2023.

Cf. A000108, A000260, A002293, A006013, A069271, (1/2)*A112385, A120588, A130564.

A006632:= func< n | Binomial(4*n-2,n-1)/n >;
[A006632(n): n in [1..40]]; // G. C. Greubel, Sep 01 2025

A006632:=n->3*binomial(4*n-1,n-1)/(4*n-1): seq(A006632(n), n=1..30); # Wesley Ivan Hurt, Oct 23 2017

InverseSeries[Series[y*(1-y)^3, {y, 0, 24}], x] (* then A(x)=y(x) *) (* Len Smiley, Apr 07 2000 *)
a[ n_] := If[n<1, 0, Binomial[4 n - 2, n - 1] / n]; (* Michael Somos, Aug 22 2014 *)

a(n) = 3*binomial(4*n-1, n-1)/(4*n-1) \\ Felix Fröhlich, Oct 23 2017

def A006632(n): return binomial(4*n-2,n-1)//n
print([A006632(n) for n in range(1,41)]) # G. C. Greubel, Sep 01 2025

a(n) = binomial(4*n-1, n)/(4*n-1) = 3*binomial(4*n-2, n-1) - binomial(4*n-2, n). - David Callan, Sep 15 2004
G.f.: g^3 where g = 1+x*g^4 is the g.f. of A002293. - Mark van Hoeij, Nov 11 2011
a(n) = (3/4)*binomial(4*n,n)/(4*n-1). - Bruno Berselli, Jan 17 2014
From Wolfdieter Lang, Feb 06 2020: (Start)
G.f.: (3/4)*(1 - hypergeometric3F2([-1, 1, 2]/4, [1, 2]/3, (4^4/3^3)*x)).
E.g.f.: (3/4)*(1 - hypergeometric3F3([-1, 1, 2]/4, [1, 2, 3]/3, (4^4/3^3)*x)). (End)
D-finite with recurrence 3*n*(3*n-1)*(3*n-2)*a(n) -8*(4*n-5)*(4*n-3)*(2*n-1)*a(n-1)=0. - R. J. Mathar, May 07 2021
a(n) = (2n-1)*A000260(n). - F. Chapoton, Jul 15 2021
G.f. A(x) satisfies: A(x) = x / (1 - A(x))^3. - Ilya Gutkovskiy, Nov 03 2021
G.f.: x*( Sum_{n >= 0} binomial(4*n+3, n)*x^n ) / ( Sum_{n >= 0} binomial(4*n, n)*x^n ) = x*( Sum_{n >= 0} binomial(4*n+3, n)*x^n ) / ( 1 + 4*x*Sum_{n >= 0} binomial(4*n+3, n)*x^n ). - Peter Bala, Dec 13 2024
Working with a offset of 0, the g.f. A(x) = 1 + 3*x + 15*x^2 + ... is uniquely determined by the conditions A(0) = 1 and [x^n] A(x)^(-n) = -3 for all n >= 1. - Peter Bala, Jul 24 2025

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

A069271 a(n) = binomial(4n+1,n)2/(3*n+2).

Original entry on oeis.org

1, 2, 9, 52, 340, 2394, 17710, 135720, 1068012, 8579560, 70068713, 580034052, 4855986044, 41043559340, 349756577100, 3001701610320, 25921837477692, 225083787458904, 1963988670706228, 17211860478150800, 151433425446423120

Offset: 0

Henry Bottomley, Mar 12 2002

nonn

This sequence counts the set B_n of plane trees defined in the Poulalhon and Schaeffer link (Definition 2.2 and Section 4.2, Proposition 4). - David Callan, Aug 20 2014
a(n) is the number of lattice paths of length 4n starting and ending on the x-axis consisting of steps {(1, 1), (1, -3)} that remain on or above the line y=-1. - Sarah Selkirk, Mar 31 2020
a(n) is the number of ordered pairs of 4-ary trees with a (summed) total of n internal nodes. - Sarah Selkirk, Mar 31 2020

			a(3) = C(4*3+1,3)*2/(3*3+2) = C(13,3)*2/11 = 286*2/11 = 52.
a(3) = 52 since the top row of M^3 = (22, 22, 7, 1).
1 + 2*x + 9*x^2 + 52*x^3 + 340*x^4 + 2394*x^5 + 17710*x^6 + 135720*x^7 + ...
q + 2*q^3 + 9*q^5 + 52*q^7 + 340*q^9 + 2394*q^11 + 17710*q^13 + 135720*q^15 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Thomas Anderson, T. Bruce McLean, Homeira Pajoohesh, and Chasen Smith , The combinatorics of all regular flexagons, Eu. J. Combinat. 31 (2010) 72-80, Theorem 2.
Roland Bacher, Fair Triangulations, arXiv:0710.0960 [math.CO], 2007.
Esther Banaian, Elise Catania, Christian Gaetz, Miranda Moore, Gregg Musiker, and Kayla Wright, Twists, Higher Dimer Covers, and Web Duality for Grassmannian Cluster Algebras, arXiv:2507.15211 [math.CO], 2025. See p. 28.
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
Gi-Sang Cheon, S.-T. Jin and L. W. Shapiro, A combinatorial equivalence relation for formal power series, Linear Algebra and its Applications, Volume 491, 15 February 2016, Pages 123-137.
Emmanuel Guitter, The distance-dependent two-point function of triangulations: a new derivation from old results, Ann. Inst. Henri Poincaré Comb. Phys. Interact. Vol. 4 (2017), 177-211. DOI: 10.4171/AIHPD/38.
Clemens Heuberger, Sarah J. Selkirk, and Stephan Wagner, Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo k, arXiv:2204.14023 [math.CO], 2022.
Ionut E. Iacob, T. Bruce McLean and Hua Wang, The V-flex, Triangle Orientation, and Catalan Numbers in Hexaflexagons, The College Mathematics Journal, Vol. 43, No. 1 (January 2012), pp. 6-10.
Jean-Christophe Novelli and Jean-Yves Thibon, Noncommutative Symmetric Functions and Lagrange Inversion, arXiv:math/0512570 [math.CO], 2005-2006.
C. O. Oakley and R. J. Wisner, Flexagons, Am. Math. Monthly 64 (3) (1957) 143-154, u_{3k+2}.
Karol A. Penson and Karol Zyczkowski, Product of Ginibre matrices : Fuss-Catalan and Raney distribution, arXiv:1103.3453 [math-ph], 2011.
Karol A. Penson and Karol Zyczkowski, Product of Ginibre matrices : Fuss-Catalan and Raney distribution, Phys. Rev. E 83, 061118, 2011.
Dominique Poulalhon and Gilles Schaeffer, Optimal Coding and Sampling of Triangulations, in Automata, Languages and Programming, Lecture Notes in Computer Science, Volume 2719, 2003, pp 1080-1094.
Sarah J. Selkirk, On a generalisation of k-Dyck paths, MSc Thesis, 2019.

Cf. A002293, A006013, A006632, A069270 for similar generalized Catalan sequences.

Cf. A000260, A115141, A163456, A224274.

[2*Binomial(4*n+1, n)/(3*n+2): n in [0..20]];  // Bruno Berselli, Mar 04 2011

BB:=[T,{T=Prod(Z,Z,Z,F,F),F=Sequence(B),B=Prod(F,F,F,Z)}, unlabeled]: seq(count(BB,size=i),i=3..23); # Zerinvary Lajos, Apr 22 2007

f[n_] := 2 Binomial[4 n + 1, n]/(3 n + 2); Array[f, 21, 0] (* Robert G. Wilson v *)

a(n)=if(n<0,0,polcoeff(serreverse(x/(1+x^2)^2+O(x^(2*n+2))),2*n+1)) /* Ralf Stephan */

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

{a(n) =  local(A); if( n<0, 0, A = 1 + O(x); for( k=1, n, A = (1 + x * A^2)^2); polcoeff( A, n))} /* Michael Somos, Mar 28 2012 */

a(n) = A069270(n+1, n) = A005810(n)*A016813(n)/A060544(n+1)
O.g.f. A(x) satisfies 2*x^2*A(x)^3 = 1-2*x*A(x)-sqrt(1-4*x*A(x)). - Vladimir Kruchinin, Feb 23 2011
a(n) is the sum of top row terms in M^n, where M is the infinite square production matrix with the triangular series in each column as follows, with the rest zeros:
   1,  1,  0, 0, 0, 0, ...
   3,  3,  1, 0, 0, 0, ...
   6,  6,  3, 1, 0, 0, ...
  10, 10,  6, 3, 1, 0, ...
  15, 15, 10, 6, 3, 1, ...
  ... - Gary W. Adamson, Aug 11 2011
Given g.f. A(x) then B(x) = x * A(x^2) satisfies x = B(x) / (1 + B(x)^2)^2. - Michael Somos, Mar 28 2012
Given g.f. A(x) then A(x) = (1 + x * A(x)^2)^2. - Michael Somos, Mar 28 2012
a(n) / (n+1) = A000260(n). - Michael Somos, Mar 28 2012
REVERT transform is A115141. - Michael Somos, Mar 28 2012
D-finite with recurrence 3*n*(3*n+2)*(3*n+1)*a(n) - 8*(4*n+1)*(2*n-1)*(4*n-1)*a(n-1) = 0. - R. J. Mathar, Jun 07 2013
a(n) = 2*binomial(4n+1,n-1)/n for n>0, a(0)=1. - Bruno Berselli, Jan 19 2014
G.f.: hypergeom([1/2, 3/4, 5/4], [4/3, 5/3], (256/27)*x). - Robert Israel, Aug 24 2014
From Peter Bala, Oct 08 2015: (Start)
O.g.f. A(x) = (1/x) * series reversion (x/C(x)^2), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. Cf. A163456.
(1/2)*x*A'(x)/A(x) is the o.g.f. for A224274. (End)
E.g.f.: hypergeom([1/2, 3/4, 5/4], [1, 4/3, 5/3], (256/27)*x). - Karol A. Penson, Jun 26 2017
a(n) = binomial(4*n+2,n)/(2*n+1). - Alexander Burstein, Nov 08 2021

A024492 Catalan numbers with odd index: a(n) = binomial(4n+2, 2n+1)/(2*n+2).

Original entry on oeis.org

1, 5, 42, 429, 4862, 58786, 742900, 9694845, 129644790, 1767263190, 24466267020, 343059613650, 4861946401452, 69533550916004, 1002242216651368, 14544636039226909, 212336130412243110, 3116285494907301262, 45950804324621742364, 680425371729975800390

Offset: 0

Clark Kimberling

a(n) and Catalan(n) have the same 2-adic valuation (equal to 1 less than the sum of the digits in the binary representation of (n + 1)). In particular, a(n) is odd iff n is of the form 2^m - 1. - Peter Bala, Aug 02 2016

			sqrt((1/2)*(1+sqrt(1-x))) = 1 - (1/8)*x - (5/128)*x^2 - (42/2048)*x^3 - ...

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, Advances in Applied Mathematics, Vol. 121 (2020), 102105; arXiv preprint, arXiv:1907.10725 [math.CO], 2019.
Pedro J. Miana and Natalia Romero, Moments of combinatorial and Catalan numbers, Journal of Number Theory, Vol. 130, No. 8 (August 2010), pp. 1876-1887. See Remark 3, p. 1882.

Cf. A048990 (Catalan numbers with even index), A024491, A000108, A000894.

Cf. A000260, A006013, A187357, A187358, A187359, A228329, A276484.

[Factorial(4*n+2)/(Factorial(2*n+1)*Factorial(2*n+2)): n in [0..20]]; // Vincenzo Librandi, Sep 13 2011

with(combstruct):bin := {B=Union(Z,Prod(B,B))}: seq (count([B,bin,unlabeled],size=2*n), n=1..18); # Zerinvary Lajos, Dec 05 2007
a := n -> binomial(4*n+1, 2*n+1)/(n+1):
seq(a(n), n=0..17); # Peter Luschny, May 30 2021

CoefficientList[ Series[1 + (HypergeometricPFQ[{3/4, 1, 5/4}, {3/2, 2}, 16 x] - 1), {x, 0, 17}], x]
CatalanNumber[Range[1,41,2]] (* Harvey P. Dale, Jul 25 2011 *)

a(n):=sum((k+1)^2*binomial(2*(n+1),n-k)^2,k,0,n)/(n+1)^2; /* Vladimir Kruchinin, Oct 14 2014 */

combinat::catalan(2*n+1)$ n = 0..24 // Zerinvary Lajos, Jul 02 2008

combinat::dyckWords::count(2*n+1)$ n = 0..24 // Zerinvary Lajos, Jul 02 2008

a(n)=binomial(4*n+2,2*n+1)/(2*n+2) \\ Charles R Greathouse IV, Sep 13 2011

G.f.: (1/2)*x^(-1)*(1-sqrt((1/2)*(1+sqrt(1-16*x)))).
G.f.: 3F2([3/4, 1, 5/4], [3/2, 2], 16*x). - Olivier Gérard, Feb 16 2011
a(n) = 4^n*binomial(2n+1/2, n)/(n+1). - Paul Barry, May 10 2005
a(n) = binomial(4n+1,2n+1)/(n+1). - Paul Barry, Nov 09 2006
a(n) = (1/(2*Pi))*Integral_{x=-2..2} (2+x)^(2*n)*sqrt((2-x)*(2+x)). - Peter Luschny, Sep 12 2011
D-finite with recurrence (n+1)*(2*n+1)*a(n) -2*(4*n-1)*(4*n+1)*a(n-1)=0. - R. J. Mathar, Nov 26 2012
G.f.: (c(sqrt(x)) - c(-sqrt(x)))/(2*sqrt(x)) = (2-(sqrt(1-4*sqrt(x)) + sqrt(1+4*sqrt(x))))/(4*x), with the g.f. c(x) of the Catalan numbers A000108. - Wolfdieter Lang, Feb 23 2014
a(n) = Sum_{k=0..n} (k+1)^2*binomial(2*(n+1),n-k)^2 /(n+1)^2. - Vladimir Kruchinin, Oct 14 2014
G.f.: A(x) = (1/x)*(inverse series of x - 5*x^2 + 8*x^3 - 4*x^4). - Vladimir Kruchinin, Oct 31 2014
a(n) ~ sqrt(2)*16^n/(sqrt(Pi)*n^(3/2)). - Ilya Gutkovskiy, Aug 02 2016
Sum_{n>=0} 1/a(n) = A276484. - Amiram Eldar, Nov 18 2020
G.f.: A(x) = C(4*x)*C(x*C(4*x)), where C(x) is the g.f. of A000108. - Alexander Burstein, May 01 2021
a(n) = (1/Pi)*16^(n+1)*Integral_{x=0..Pi/2} (cos x)^(4n+2)*(sin x)^2. - Greg Dresden, May 30 2021
Sum_{n>=0} a(n)/4^n = 2 - sqrt(2). - Amiram Eldar, Mar 16 2022
From Peter Bala, Feb 22 2023: (Start)
a(n) = (1/4^n) * Product_{1 <= i <= j <= 2*n} (i + j + 2)/(i + j - 1).
a(n) = Product_{1 <= i <= j <= 2*n} (3*i + j + 2)/(3*i + j - 1). Cf. A000260. (End)

More terms from Wolfdieter Lang

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

cp's OEIS Frontend

A362102 Erroneous version of A000260.

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A002293 Number of dissections of a polygon: binomial(4*n, n)/(3*n + 1).

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A006013 a(n) = binomial(3*n+1,n)/(n+1).

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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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A006632 a(n) = 3*binomial(4*n-1, n-1)/(4*n-1).

A002293 Number of dissections of a polygon: binomial(4n, n)/(3n + 1).

A006632 a(n) = 3binomial(4n-1, n-1)/(4*n-1).

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

A069271 a(n) = binomial(4n+1,n)2/(3*n+2).

A024492 Catalan numbers with odd index: a(n) = binomial(4n+2, 2n+1)/(2*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).