A000087 -id:A000087 - cp's OEIS frontend

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

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.
F. Chapoton and S. Giraudo, Enveloping operads and bicoloured noncrossing configurations, arXiv:1310.4521 [math.CO], 2013.
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.
Emeric Deutsch, S. Feretic and M. Noy, Diagonally convex directed polyominoes and even trees: a bijection and related issues, Discrete Math., 256 (2002), 645-654.
I. Gessel and G. Xin, The generating function of ternary trees and continued fractions, arXiv:math/0505217 [math.CO], 2005.
Samuele Giraudo, Tree series and pattern avoidance in syntax trees, arXiv:1903.00677 [math.CO], 2019.
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.
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.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 432 [broken link].
Pakawut Jiradilok, Large-scale Rook Placements, arXiv:2204.00615 [math.CO], 2022.
S. Kitaev and A. de Mier, Enumeration of fixed points of an involution on beta(1, 0)-trees, arXiv:1210.2618 [math.CO], 2012. - From _N. J. A. Sloane_, Dec 31 2012
Sergey Kitaev, Anna de Mier, and Marc Noy, On the number of self-dual rooted maps, European J. Combin. 35 (2014), 377-387. MR3090510. See Theorem 1. - _N. J. A. Sloane_, May 19 2014
Don Knuth, 20th Anniversary Christmas Tree Lecture.
Philippe Leroux, An algebraic framework of weighted directed graphs, Int. J. Math. Math. Sci. 58. (2003).
Philippe Leroux, L-algebras, triplicial-algebras, within an equivalence of categories motivated by graphs, arXiv:0709.3453 [math-ph], 2008.
Ho-Hon Leung and Thotsaporn "Aek" Thanatipanonda, A Probabilistic Two-Pile Game, arXiv:1903.03274 [math.CO], 2019.
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
Hugo Mlodecki, Decompositions of packed words and self duality of Word Quasisymmetric Functions, arXiv:2205.13949 [math.CO], 2022. See Table 4 p. 20.
W. Mlotkowski and K. A. Penson, The probability measure corresponding to 2-plane trees, arXiv:1304.6544 [math.PR], 2013.
Henri Muehle and Philippe Nadeau, A Poset Structure on the Alternating Group Generated by 3-Cycles, arXiv:1803.00540 [math.CO], 2018.
Liviu I. Nicolaescu, Counting Morse functions on the 2-sphere, arXiv:math/0512496 [math.GT], 2005.
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv:1403.5962 [math.CO], 2014.
M. Noy, Enumeration of noncrossing trees on a circle, Discrete Math., 180, 301-313, 1998.
Isaac Owino Okoth, Bijections of k-plane trees, Open J. Discret. Appl. Math. (2022) Vol. 5, No. 1, 29-35.
J.-B. Priez and A. Virmaux, Non-commutative Frobenius characteristic of generalized parking functions: Application to enumeration, arXiv:1411.4161 [math.CO], 2014-2015.
Helmut Prodinger, On some questions by Cameron about ternary paths --- a linear algebra approach, arXiv:1910.02320 [math.CO], 2019.
Helmut Prodinger, Sarah J. Selkirk, and Stephan Wagner, On two subclasses of Motzkin paths and their relation to ternary trees, arXiv:1902.01681 [math.CO], 2019.
Jocelyn Quaintance, Combinatoric Enumeration of Two-Dimensional Proper Arrays, Discrete Math., 307 (2007), 1844-1864.
Thomas M. Richardson, The three 'R's and Dual Riordan Arrays, arXiv:1609.01193 [math.CO], 2016.
D. G. Rogers, Comments on A111160, A055113 and A006013.
Joe Sawada, Jackson Sears, Andrew Trautrim, and Aaron Williams, Demystifying our Grandparent's De Bruijn Sequences with Concatenation Trees, arXiv:2308.12405 [math.CO], 2023.
Michael Somos, Number Walls in Combinatorics.
Jun Yan, Lattice paths enumerations weighted by ascent lengths, arXiv:2501.01152 [math.CO], 2025. See p. 7.
Zhujun Zhang, A Note on Counting Dependency Trees, arXiv:1708.08789 [math.GM], 2017. See p. 3.
S.-n. Zheng and S.-l. Yang, On the-Shifted Central Coefficients of Riordan Matrices, Journal of Applied Mathematics, Volume 2014, Article ID 848374, 8 pages.

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

A006402 Number of sensed 2-connected (nonseparable) planar maps with n edges.

Original entry on oeis.org

1, 2, 3, 6, 16, 42, 151, 596, 2605, 12098, 59166, 297684, 1538590, 8109078, 43476751, 236474942, 1302680941, 7256842362, 40832979283, 231838418310, 1327095781740, 7653155567834, 44434752082990, 259600430870176, 1525366978752096, 9010312253993072, 53485145730576790

Offset: 2

N. J. A. Sloane

nonn

Some people begin this 2,1,2,3,6,..., others begin it 0,1,2,3,6,....

The dual of a nonseparable map is nonseparable, so the class of all nonseparable planar maps is self-dual. The maps considered here are unrooted and sensed and may include loops and parallel edges. - Andrew Howroyd, Mar 29 2021

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
T. R. S. Walsh, personal communication.

Andrew Howroyd, Table of n, a(n) for n = 2..500
V. A. Liskovets, T. R. S. Walsh, The enumeration of nonisomorphic 2-connected planar maps, Canad. J. Math. 35 (1983), no. 3, 417-435.
Timothy R. Walsh, Generating nonisomorphic maps without storing them, SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 161-178.
T. R. S. Walsh, Number of sensed planar maps with n edges and m vertices

Row sums of A342061.

Cf. A000087 (with distinguished faces), A000139 (rooted), A005645, A006403 (unsensed), A006406 (without loops or parallel edges).

\\ here r(n) is A000139(n-1).
r(n)={4*binomial(3*n,n)/(3*(3*n-2)*(3*n-1))}
a(n)={(r(n) + sumdiv(n, d, if(dAndrew Howroyd, Mar 29 2021

Terms a(23) and beyond from Andrew Howroyd, Mar 29 2021

A046646 a(n) = 2binomial(3n-3, n-1)/(2*n-1) for n >= 2, and a(1) = 1.

Original entry on oeis.org

1, 2, 6, 24, 110, 546, 2856, 15504, 86526, 493350, 2861430, 16829280, 100134216, 601661144, 3645533040, 22249511328, 136657509918, 844061090670, 5239262085330, 32665844580600, 204480219795390, 1284624902435490

Offset: 1

N. J. A. Sloane

Number of certain rooted planar maps.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Daniel Birmajer, Juan B. Gil, Michael D. Weiner, Bounce statistics for rational lattice paths, arXiv:1707.09918 [math.CO], 2017, p. 10.
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]

A diagonal of A046651.

Cf. A046647, A005470, A000139, A001764.

[1] cat [2*Binomial(3*n-3,n-1)/(2*n-1): n in [2..30]]; // Vincenzo Librandi, Oct 13 2013

alias(PS=ListTools:-PartialSums): A046646List := proc(m) local A, P, n;
A := [1,2]; P := [2]; for n from 1 to m - 2 do P := PS(PS([op(P), P[-1]]));
A := [op(A), P[-1]] od; A end: A046646List(22); # Peter Luschny, Mar 26 2022

Join[{1},Table[(2*Binomial[3n-3,n-1])/(2n-1),{n,2,30}]] (* Harvey P. Dale, Oct 12 2013 *)

From Emeric Deutsch, Mar 03 2004: (Start)
a(n) = 2*binomial(3*n-3, n-1)/(2*n-1) for n >= 2, and a(1) = 1.
a(n) = 2*A001764(n-1) for n >= 2. (End)
a(n) = (n+1) * A000139(n). - F. Chapoton, Feb 23 2024
G.f.: (1+g)/(1-g) where g*(1-g)^2 = x. - Mark van Hoeij, Nov 10 2011

More terms from Emeric Deutsch, Mar 03 2004

New name using a formula of Emeric Deutsch by Peter Luschny, Feb 23 2024

A046651 Triangle of rooted planar maps.

Original entry on oeis.org

1, 1, 2, 1, 6, 6, 1, 12, 26, 24, 1, 20, 75, 120, 110, 1, 30, 174, 416, 594, 546, 1, 42, 350, 1176, 2289, 3094, 2856, 1, 56, 636, 2880, 7322, 12768, 16728, 15504, 1, 72, 1071, 6324, 20475, 44388, 72420, 93024, 86526, 1, 90, 1700, 12740, 51495, 136252, 267240, 417240, 528770, 493350, 1, 110, 2574, 23936, 118734, 378444, 878460, 1610136, 2437149, 3058770, 2861430

Offset: 0

N. J. A. Sloane

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

A091599 is the same triangle with rows reversed and has much more information.

T := proc(n, k) if k<=n then k*sum((2*j-k)*(j-1)!*(3*n-j-k-1)!/(j-k)!/(j-k)!/(2*k-j)!/(n-j)!, j=k..min(n, 2*k))/(2*n-k)! else 0 fi end: seq(seq(T(n, n-k+1), k=1..n), n=1..11); # Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 19 2008

t[n_, k_] := If[k <= n, k*Sum[(2*j-k)*(j-1)!*(3*n-j-k-1)!/(j-k)!/(j-k)!/(2*k-j)!/(n-j)!, {j, k, Min[n, 2*k]}]/(2*n-k)!, 0]; Table[Table[t[n, n-k+1], {k, 1, n}], {n, 1, 11}] // Flatten (* Jean-François Alcover, Jan 14 2014, after Herman Jamke *)

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 19 2008

A046653 Triangle of rooted planar maps up to orientation-preserving isomorphisms.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 3, 4, 1, 2, 8, 12, 14, 1, 3, 12, 30, 43, 49, 1, 3, 22, 63, 134, 189, 216, 1, 4, 31, 133, 323, 608, 888, 984, 1, 4, 48, 238, 759, 1671, 2988, 4332, 4862, 1, 5, 64, 422, 1594, 4260, 8772, 15126, 22011, 24739, 1, 5, 90, 694, 3210, 10002, 23986, 46923, 79214, 115131, 130338, 1, 6, 115, 1106, 6024, 22176, 60813, 135097, 255277, 424034, 616395, 701584

Offset: 2

N. J. A. Sloane

			Triangle begins:
  1;
  1,  1;
  1,  1,  2;
  1,  2,  3,  4;
  1,  2,  8, 12, 14;
  1,  3, 12, 30, 43, 49;
  ...

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]

Cf. A046650 (diagonal), A000087 (row sums).

# uses function L(.,.) from A046650
for n from 2 to 13 do
for m from n to 2 by -1 do
    printf("%d,",L(n,m)) ;
end do:
printf("\n") ;
end do: # R. J. Mathar, Apr 13 2019

Reference gives generating functions.

More terms from R. J. Mathar, Apr 13 2019

A046652 Triangle of rooted planar maps, read by rows.

Original entry on oeis.org

1, 2, 2, 3, 8, 7, 4, 21, 34, 30, 5, 44, 114, 160, 143, 6, 80, 308, 609, 806, 728, 7, 132, 715, 1908, 3315, 4256, 3876, 8, 203, 1482, 5185, 11420, 18444, 23256, 21318, 9, 296, 2814, 12600, 34520, 67856, 104652, 130416, 120175, 10, 414, 4984, 27965, 93924, 221300, 404016, 603801, 746350, 690690, 11, 560, 8343, 57584, 234066, 654336, 1394505, 2418372, 3533145, 4341480, 4032015

Offset: 0

N. J. A. Sloane

			Triangle begins:
  1;
  2,   2;
  3,   8,   7;
  4,  21,  34,  30;
  5,  44, 114, 160, 143;
  6,  80, 308, 609, 806, 728;
  ...

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]

A091665 is the same triangle with rows reversed and has much more information.

T := proc(n, k) if k<=n then k*sum((2*j-k+1)*(j-1)!*(3*n-k-j)!/(j-k+1)!/(j-k)!/(2*k-j-1)!/(n-j)!, j=k..min(n, 2*k-1))/(2*n-k+1)! else 0 fi end: seq(seq(T(n, n-k+1), k=1..n), n=1..11); # Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 30 2008

t[n_, k_] := If[k <= n, k*Sum[(2*j-k+1)*(j-1)!*(3*n-k-j)!/(j-k+1)!/(j-k)!/(2*k-j-1)!/(n-j)!, {j, k, Min[n, 2*k-1]}]/(2*n-k+1)!, 0]; Table[t[n, k], {n, 1, 11}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jan 20 2014, after Herman Jamke *)

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 30 2008

A000259 Number of certain rooted planar maps.

Original entry on oeis.org

1, 3, 13, 63, 326, 1761, 9808, 55895, 324301, 1908878, 11369744, 68395917, 414927215, 2535523154, 15592255913, 96419104103, 599176447614, 3739845108057, 23435007764606, 147374772979438, 929790132901804, 5883377105975922, 37328490926964481, 237427707464042693

Offset: 1

N. J. A. Sloane

nonn

a(n) is also the number of North-East lattice paths from (0,0) to (2n,n) that begin with a North step, end with an East step, and do not bounce off the line y =1/2 x. Similarly, also the number of paths that begin with an East step, end with a North step, and do not bounce off the line y=1/2 x. - Michael D. Weiner, Aug 03 2017

			For n = 2 the a(2) = 3 is counting the following three paths EEEENN, EEENEN, ENEEEN. The path EENEEN is excluded as it bounces off the line y = (1/2) x at the point (2, 1). - _Michael D. Weiner_, Aug 03 2017

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).

Michael De Vlieger, Table of n, a(n) for n = 1..1211
D. Birmajer, J. Gil, and M. Weiner, Bounce statistics for rational lattice paths, arXiv:1707.09918 [math.CO], 2017.
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]
Anthony G. Shannon, Hakan Akkuş, Yeşim Aküzüm, Ömür Deveci, and Engin Özkan, A partial recurrence Fibonacci link, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 530-537. See Table 2, p. 534.

Row sums of A046651.

[&+[(-1)^(k-1)*Binomial(3*n, n-k)*k/n*Fibonacci(k - 2):k in [0..n]]: n in [1..30]]; // Vincenzo Librandi, Aug 05 2017

with(linalg): T := proc(n,k) if k<=n then k*add((2*j-k)*(j-1)!*(3*n-j-k-1)!/(j-k)!/(j-k)!/(2*k-j)!/(n-j)!,j=k..min(n,2*k))/(2*n-k)! else 0 fi end: A := matrix(30,30,T): seq(add(A[i,j],j=1..i),i=1..30); # Emeric Deutsch, Mar 03 2004
R := RootOf(x-t*(t-1)^2, t); ogf := series(1/((1-R-R^2)*(R-1)^2), x=0, 30); # Mark van Hoeij, Nov 08 2011

R = Root[#^3-2#^2+#-x&, 1]; CoefficientList[1/((1-R-R^2)*(R-1)^2) + O[x]^30, x] (* Jean-François Alcover, Feb 06 2016, after Mark van Hoeij *)
Table[Sum[(-1)^(k - 1)*Binomial[3 n, n - k]*k/n*Fibonacci[k - 2], {k, n}], {n, 21}] (* Michael De Vlieger, Aug 04 2017 *)

a(n) = sum(k = 1, n, (-1)^(k-1)*binomial(3*n,n-k)*k/n*fibonacci(k-2)); \\ Michel Marcus, Aug 04 2017

from sympy import binomial, fibonacci
def a(n): return sum((-1)**(k - 1)*binomial(3*n, n - k)*k//n*fibonacci(k - 2) for k in range(1, n + 1))
print([a(n) for n in range(1, 21)]) # Indranil Ghosh, Aug 05 2017

a(n) = Sum_{k = 1..n} (-1)^(k-1)*C(3n, n-k)*k/n*F(k-2) where F(k) is the k-th Fibonacci number (A000045) and F(-1) = 1. - Michael D. Weiner, Aug 03 2017

a(n) ~ 3^(3*n + 1/2) / (5 * sqrt(Pi) * n^(3/2) * 2^(2*n - 1)). - Vaclav Kotesovec, Jul 06 2024

More terms from Emeric Deutsch, Mar 03 2004

A000305 Number of certain rooted planar maps.

Original entry on oeis.org

1, 4, 18, 89, 466, 2537, 14209, 81316, 473338, 2793454, 16674417, 100487896, 610549829, 3735850007, 23000055178, 142370597601, 885521350882, 5531501612071, 34686798239678, 218273864005214, 1377897874711437

Offset: 1

N. J. A. Sloane

nonn

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 = 1..200
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]

Row sums of A046652.

with(linalg): T := proc(n,k) if k<=n then k*sum((2*j-k+1)*(j-1)!*(3*n-k-j)!/(j-k+1)!/(j-k)!/(2*k-j-1)!/(n-j)!,j=k..min(n,2*k-1))/(2*n-k+1)! else 0 fi end:A := matrix(30,30,T): seq(sum(A[i,j],j=1..i),i=1..30);
R := RootOf(x-t*(t-1)^2, t); ogf := series((R+1)/((1-R-R^2)*(R-1)^2), x=0, 20); # Mark van Hoeij, Nov 08 2011

t[n_, k_] := If[k <= n, k*Sum[(2*j-k+1)*(j-1)!*(3*n-k-j)!/(j-k+1)!/(j-k)!/ (2*k-j-1)!/(n-j)!, {j, k, Min[n, 2*k-1]}]/(2*n-k+1)!, 0]; a[n_] := Sum[ t[n, k], {k, 1, n}]; Array[a, 21] (* Jean-François Alcover, Feb 07 2016 after Herman Jamke in A046652 *)

More terms from Emeric Deutsch, Mar 03 2004

A046647 Number of certain rooted planar maps.

Original entry on oeis.org

1, 6, 26, 120, 594, 3094, 16728, 93024, 528770, 3058770, 17948970, 106585440, 639318456, 3867821640, 23574446992, 144621823632, 892293152994, 5533289372170, 34468829508750, 215594574231960, 1353464311979010

Offset: 2

N. J. A. Sloane

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]

A diagonal of A046651. Cf. A046646.

a(2)=1; a(n) = 4*(7*n-15)*(3*n-6)!/((n-2)!*(2n-2)!). - Emeric Deutsch, Mar 03 2004

G.f.: (g+1)*(3*g+1)/(g-1)^2 where g*(1-g)^2 = x. - Mark van Hoeij, Nov 10 2011

More terms from Emeric Deutsch, Mar 03 2004

A046650 Number of sensed nonseparable (2-connected) planar maps with n edges and a distinguished face of size 2.

Original entry on oeis.org

1, 1, 2, 4, 14, 49, 216, 984, 4862, 24739, 130338, 701584, 3852744, 21489836, 121525520, 695307888, 4019381790, 23446201495, 137875564710, 816646459860, 4868578092510, 29196022525905, 176022392938080, 1066433501134560, 6490009570139784, 39659537885087124, 243278423033093336, 1497584057249141728, 9249144367260811824

Offset: 2

N. J. A. Sloane

From R. J. Mathar, Apr 13 2019: (Start)
Table III with row sums A000087 is (A046653 row-reversed):
     1;
     1,    1;
     2,    1,    1;
     4,    3,    2,    1;
    14,   12,    8,    2,   1;
    49,   43,   30,   12,   3,   1;
   216,  189,  134,   63,  22,   3,  1;
   984,  888,  608,  323, 133,  31,  4, 1;
  4862, 4332, 2988, 1671, 759, 238, 48, 4, 1;
  ...
(End)
Equivalently, by duality, a(n) is the number of sensed nonseparable planar maps with n edges and a distinguished vertex of degree 2.  - Andrew Howroyd, Jan 19 2025

Andrew Howroyd, Table of n, a(n) for n = 2..500
W. G. Brown, Enumeration of non-separable planar maps, Canad. J. Math., 15 (1963), 526-545.

Main diagonal of A046653.

Cf. A000087 (distinguished face of any size).

B1nm := proc(n,m) # eq (4.15)
    local j ;
    if m>=2 and n>= m  then
        add((3*m-2*j-1)*(2*j-m)*(j-2)!*(3*n-j-m-1)!/(n-j)!/(j-m)!/(j-m+1)!/(2*m-j)!,j=m..min(n,2*m) ) ;
        %*m/(2*n-m)! ;
    else
        0 ;
    end if;
end proc:
B2wj := proc(w,j) # eq (8.21)
    local k ;
    if  w >= j and j>=1 and w >= 1 then
        add((2*k-j+1)*(k-1)!*(3*w-k-j)!/(k-j+1)!/(k-j)!/(2*j-k-1)!/(w-k)!,k=j..min(w,2*j-1) ) ;
        %*j/(2*w-j+1)! ;
    else
        0;
    end if;
end proc:
Brwj := proc(r,w,j) # eq. (8.21)
    local k ;
    if  w >= j and j>=1 and w>=1 and r > 1 then
        add((2*k-j)*(k-1)!*(3*w-k-j-1)!/((k-j)!)^2/(2*j-k)!/(w-k)!,k=j..min(w,2*j) ) ;
        %*j/(2*w-j)! ;
    else
        0 ;
    end if;
end proc:
Brnm := proc(r,n,m)
    if r = 1 then
        B1nm(n,m) ;
    elif r = 2 and type(n,'odd') and type (m,'even') then
        B2wj((n-1)/2,m/2) ;
    elif modp(n,r) <> 0 or modp(m,r) <> 0 then
        0;
    else
        Brwj(r,n/r,m/r) ;
    end if;
end proc:
L := proc(n,m) # eq. (6.7)
    add(numtheory[phi](s)*Brnm(s,n,m),s=numtheory[divisors](m)) ;
    %/m ;
end proc:
seq(L(n,2),n=2..40) ; # R. J. Mathar, Apr 13 2019

B1nm[n_, m_] := If[m >= 2 && n >= m, Sum[(3m - 2j - 1)(2j - m)(j - 2)! (3n - j - m - 1)!/(n - j)!/(j - m)!/(j - m + 1)!/(2m - j)!, {j, m, Min[n, 2m] }] m/(2n - m)!, 0];
B2wj[w_, j_]  := If[w >= j && j >= 1 && w >= 1, Sum[(2k - j + 1)(k - 1)! (3 w - k - j)!/(k - j + 1)!/(k - j)!/(2j - k - 1)!/(w - k)!, {k, j, Min[w, 2 j - 1] }] j/(2w - j + 1)!, 0];
Brwj[r_, w_, j_] := If[w >= j && j >= 1 && w >= 1 && r > 1 , Sum[(2k - j)(k - 1)! (3w - k - j - 1)!/((k - j)!)^2/(2j - k)!/(w - k)!, {k, j, Min[w, 2j]}] j/(2w - j)!, 0];
Brnm[r_, n_, m_] := Which[r == 1, B1nm[n, m], r == 2 && OddQ[n] && EvenQ[m], B2wj[(n - 1)/2, m/2], Mod[n, r] != 0 || Mod[m, r] != 0, 0, True, Brwj[r, n/r, m/r]];
L[n_, m_] := Sum[EulerPhi[s] Brnm[s, n, m], {s, Divisors[m]}]/m;
Table[L[n, 2], {n, 2, 30}] // Flatten (* Jean-François Alcover, Apr 05 2020, after R. J. Mathar *)

a(n) = n-=2; if(n<=0, n==0, (n*binomial(3*n\2, n\2) + binomial(3*n, 2*n+1))/(n*(n+1)) ) \\ Andrew Howroyd, Jan 19 2025

Reference gives generating functions.

a(n+2) = binomial(floor(3*n/2), floor(n/2))/(n+1) + binomial(3*n, 2*n+1)/(n*(n+1)) for n > 0. - Andrew Howroyd, Jan 19 2025

More terms from R. J. Mathar, Apr 13 2019

Name clarified by Andrew Howroyd, Jan 19 2025

cp's OEIS Frontend

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

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A006402 Number of sensed 2-connected (nonseparable) planar maps with n edges.

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A046646 a(n) = 2*binomial(3*n-3, n-1)/(2*n-1) for n >= 2, and a(1) = 1.

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A046651 Triangle of rooted planar maps.

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A046653 Triangle of rooted planar maps up to orientation-preserving isomorphisms.

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A046652 Triangle of rooted planar maps, read by rows.

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A000259 Number of certain rooted planar maps.

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A046646 a(n) = 2binomial(3n-3, n-1)/(2*n-1) for n >= 2, and a(1) = 1.