A260039 -id:A260039 - cp's OEIS frontend

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

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

Offset: 0

N. J. A. Sloane

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

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

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

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

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

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

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

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

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

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

More terms from Joerg Arndt, Feb 26 2014

A005568 Product of two successive Catalan numbers C(n)*C(n+1).

Original entry on oeis.org

1, 2, 10, 70, 588, 5544, 56628, 613470, 6952660, 81662152, 987369656, 12228193432, 154532114800, 1986841476000, 25928281261800, 342787130211150, 4583937702039300, 61923368957373000, 844113292629453000, 11600528392993339800, 160599522947154548400, 2238236829690383152800

Offset: 0

R. K. Guy, Simon Plouffe and N. J. A. Sloane

Also equal to the number of standard tableaux of 2n cells with height less than or equal to 4. A005817(2n) - Mike Zabrocki, Feb 22 2007
Also equal to Sum binomial(2n,2i)*C(i)*C(n-i) = (4/Pi^2) Integral_{y=0..Pi} Integral_{x=0..Pi} (2*cos(x)+2*cos(y))^(2n)*sin^2(x)*sin^2(y) dx dy, since this counts walks of 2n steps in the nonnegative quadrant of an integer lattice that return to the origin (cf. R. K. Guy link below). - Andrew V. Sutherland, Nov 29 2007
Also, number of walks within N^2 (the first quadrant of Z^2) starting at (0,0), ending on the vertical axis and consisting of 2 n steps taken from {(-1, 0), (-1, 1), (1, -1), (1, 0)}. - Manuel Kauers, Nov 18 2008 - Manuel Kauers, Nov 18 2008
Also, number of walks within N^2 (the first quadrant of Z^2) starting and ending at (0,0) and consisting of 2 n steps taken from {(-1, 0), (0, -1), (0, 1), (1, 0)}. - Manuel Kauers, Nov 18 2008
a(2n-2) is also the sum of the numbers of standard Young tableaux of size 2n of (2,2) rectangular hook shapes (k+2,k+2,2^{n-2-k}, 0 <= k <= n-2. - Amitai Regev (amitai.regev(AT)weizmann.ac.il), Mar 10 2010
Also, number of tree-rooted planar maps with n edges. - Noam Zeilberger, Aug 18 2017

M. Lothaire, Applied Combinatorics on Words, Cambridge, 2005. See Prop. 9.1.9, p. 452. [From N. J. A. Sloane, Apr 03 2012]
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Muniru A Asiru, Table of n, a(n) for n = 0..831 (terms n=0..100 from T. D. Noe)
M. Agiorgousis, B. Green, N. A. Scoville, A. Onderdonk, and K. Rich, Homological sequences in discrete Morse theory, J. Integer Seqs., (submitted), 2012. - From _N. J. A. Sloane_, Dec 27 2012
Andrei Asinowski, Cyril Banderier, and Sarah J. Selkirk, From Kreweras to Gessel: A walk through patterns in the quarter plane, Séminaire Lotharingien de Combinatoire, Proc. 35th Conf. Formal Power Series and Alg. Comb. (Davis, 2023) Vol. 89B, Art. #30.
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
Matteo Bellitti, Siddhardh Morampudi, and Chris R. Laumann, Hamiltonian Dynamics of a Sum of Interacting Random Matrices, arXiv:1908.02263 [cond-mat.stat-mech], 2019.
Olivier Bernardi, Bijective Counting of Tree-Rooted Maps and Shuffles of Parenthesis Systems, Electronic Journal of Combinatorics, Vol. 14 (2007), Article R9.
Alin Bostan and Manuel Kauers, Automatic Classification of Restricted Lattice Walks, arXiv:0811.2899 [math.CO], 2008.
Mireille Bousquet-Mélou and Marni Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008-2009.
Steve Butler et al., Paperclip graphs, see p. 10.
Robert Cori, Serge Dulucq and Gérard Viennot, Shuffle of parenthesis systems and Baxter permutations, J. Combin. Theory, Series A, Vol. 43, No. 1 (1986), pp. 1-22.
Serge Dulucq and Olivier Guibert, Baxter permutations, Discrete Math., Vol. 180, No. 1-3 (1998), pp. 143--156. MR1603713 (99c:05004). See Th. 6.
Dominique Gouyou-Beauchamps, Chemins sous-diagonaux et tableaux de Young, pp. 112-125 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, Springer, 1986. (Annotated scanned copy)
Dominique Gouyou-Beauchamps, Standard Young tableaux of height 4 and 5, Europ. J. Combinatorics, Vol. 10, No. 1 (1989), pp. 69-82.
Dominique Gouyou-Beauchamps, Chemins sous-diagonaux et tableaux de Young, pp. 112-125 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986, p. 118 Corr. a).
Richard K. Guy, Letter to N. J. A. Sloane, May 1990.
Richard K. Guy, Letter to N. J. A. Sloane with attachment, Jun. 1991.
Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article 00.1.6.
Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2008-2010.
Gilles Lachaud, On the distribution of the trace in the unitary symplectic group and the distribution of Frobenius, arXiv preprint arXiv:1506.06482 [math.AG], 2015.
Cécilia Lancien, Patrick Oliveira Santos, and Pierre Youssef, Central Limit Theorem for tensor products of free variables, arXiv:2404.19662 [math.PR], 2024. See p. 7.
James Mallos, A 6-Letter 'DNA' for Baskets with Handles, Mathematics, Vol. 7, No. 2 (2019), Article 165.
Ronald C. Mullin, On the average activity of a spanning tree of a rooted map, J. Combin. Theory, Vol. 3, No. 2 (1967), pp. 103-121.
Ronald C. Mullin, On the average activity of a spanning tree of a rooted map, J. Combin. Theory, Vol. 3, No. 2 (1967), pp. 103-121. [Annotated scanned copy]
Liviu I. Nicolaescu, Counting Morse functions on the 2-sphere, arXiv:math/0512496 [math.GT], 2005-2006.
Justin Offutt, Automatic Bounds on Constant Term Sequences Modulo Primes, arXiv:2504.19031 [math.NT], 2025. See p. 2.
Dov Tamari, Monoïdes préordonnés et chaînes de Malcev, Bulletin de la Société Mathématique de France, Volume 82 (1954), pp. 53-96. See end of Appendix II.
Timothy R. S. Walsh and Alfred B. Lehman, Counting rooted maps by genus. III: Nonseparable maps, J. Combinatorial Theory, Ser. B, Vol. 18, No. 3 (1975), pp. 222-259: the number of parenthesis-bracket systems with n pairs.

Cf. A000108, A000356, A005817.

Cf. A004304, A168450, A168451, A168452. - Paul D. Hanna, Nov 26 2009

List([0..21],n->Binomial(2*n,n)*Binomial(2*(n+1),n+1)/((n+1)*(n+2))); # Muniru A Asiru, Dec 13 2018

[Catalan(n)*Catalan(n+1): n in [0..21]]; // Vincenzo Librandi, Feb 06 2020

A000108:=n->binomial(2*n,n)/(n+1):
seq(A000108(n)*A000108(n+1),n=0..21); # Emeric Deutsch, Mar 05 2007

f[n_] := CatalanNumber[n] CatalanNumber[n + 1] (* Or *) (4n + 2) Binomial[2 n, n]^2/(n^3 + 4n^2 +5n + 2) (* Or *) (2 n)! (2 + 2 n)!/(n! ((1 + n)!)^2 (2 + n)!); Array[f, 22, 0] (* Robert G. Wilson v *)
Times@@@Partition[CatalanNumber[Range[0,30]],2,1] (* Harvey P. Dale, Jul 23 2012 *)

(alias(C,binomial));a(n)=(C(2*n,n)-C(2*n,n-1))*(C(2*n+2,n+1)-C(2*n+2,n)) /* Michael Somos, Jun 22 2005 */

[catalan_number(i)*catalan_number(i+1) for i in range(0,22)] # Zerinvary Lajos, May 17 2009

a(n) = binomial(2*n,n)*binomial(2*n+2,n+1)/((n+1)(n+2)).
a(n) = 2*(2*n+1)*binomial(2*n,n)^2/((n+2)(n+1)^2).
D-finite with recurrence (n+2)*(n+1)*a(n) = 4*(2*n-1)*(2*n+1)*a(n-1). - Corrected R. J. Mathar, Feb 05 2020
G.f. in Maple notation: (1/2)/x+1/768/(x^2*Pi)*((32-512*x)*EllipticK(4*x^(1/2))+(-32-512*x)*EllipticE(4*x^(1/2))). - Karol A. Penson, Oct 24 2003
G.f.: 3F2( (1, 1/2, 3/2); (2, 3))(16*x) = (1 - 2F1((-1/2, 1/2); (2))( 16*x))/(2*x). - Olivier Gérard Feb 16 2011
G.f.: (1/(6*x))*(3+(16*x-1)*(2*hypergeom([1/2, 1/2],[1],16*x) + (16*x+1)*hypergeom([3/2, 3/2],[2],16*x))). - Mark van Hoeij, Nov 02 2009
G.f.: (1-hypergeom([-1/2,1/2],[2],16*x))/(2*x). - Mark van Hoeij, Aug 14 2014
E.g.f.: (1/3)*(8*x^2*BesselI(0, 2*x)^2 - 4*BesselI(0, 2*x)*BesselI(1, 2*x)*x - BesselI(1, 2*x)^2 - 8*BesselI(1, 2*x)^2*x^2)/x. - Vladeta Jovovic, Dec 29 2003
E.g.f. Sum_{n>=0} a(n)*x^(2n)/(2n)! = BesselI(1, 2x)^2/x^2. - Michael Somos, Jun 22 2005
From Paul D. Hanna, Nov 26 2009: (Start)
G.f.: A(x) = [(1/x)*Series_Reversion(x/F(x)^2)]^(1/2) where F(x) = g.f. of A004304, where A004304(n) is the number of nonseparable planar tree-rooted maps with n edges.
G.f.: A(x) = F(x*A(x)^2) where A(x/F(x)^2) = F(x) where F(x) = g.f. of A004304.
G.f.: A(x) = G(x*A(x)) where A(x/G(x)) = G(x) where G(x) = g.f. of A168450.
G.f.: A(x) = (1/x)*Series_Reversion(x/G(x)) where G(x) = g.f. of A168450.
Self-convolution yields A168452.
(End)
Representation of a(n) as the n-th power moment of a positive function on the segment [0,16]; in Mathematica notation, a(n) = NIntegrate[x^n*(8 ((1+x/16)*EllipticE[1-x/16]-1/8*x*EllipticK[1-x/16]))/(3*(Pi^2)*Sqrt[x]),{x,0,16}]. This solution of the Hausdorff power moment problem is unique. - Karol A. Penson, Oct 05 2011
G.f. y=A(x) satisfies: 0 = x^2*(16*x-1)*y''' + 6*x*(16*x-1)*y'' + 6*(18*x-1)*y' + 12*y. - Gheorghe Coserea, Jun 14 2018
Sum_{n>=0} a(n)/4^(2*n+1) = 2 - 16/(3*Pi). - Amiram Eldar, Apr 02 2022

More terms from Emeric Deutsch, Feb 20 2004
More terms from Manuel Kauers, Nov 18 2008
Two hypergeometric g.f.s, van Hoeij's formula checked and formula field edited by Olivier Gérard, Feb 16 2011

A046715 Secondary root edges in doubly rooted tree maps with n edges.

Original entry on oeis.org

0, 1, 10, 105, 1176, 13860, 169884, 2147145, 27810640, 367479684, 4936848280, 67255063876, 927192688800, 12914469594000, 181497968832600, 2570903476583625, 36671501616314400, 526348636137670500, 7597019633665077000, 110205019733436728100

Offset: 0

N. J. A. Sloane

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

B:= n-> (2*n)!*(2*n+2)!*n / (2*n!*(n+1)!^2*(n+2)!):
seq(B(n), n=0..20); # Alois P. Heinz, Dec 22 2011

B(n) = (2*n)!*(2*n+2)!*n / (2*n!*(n+1)!^2*(n+2)!). - Alois P. Heinz, Dec 22 2011

Corrected and extended by Alois P. Heinz, Dec 22 2011

A260040 Triangle read by rows giving numbers H(n,k), number of classes of twin-tree-rooted maps with n edges whose root bond contains k edges.

Original entry on oeis.org

1, 8, 1, 72, 15, 1, 720, 190, 24, 1, 7780, 2345, 415, 35, 1, 89040, 29127, 6384, 798, 48, 1, 1064644, 367248, 93324, 15162, 1400, 63, 1, 13173216, 4708344, 1332528, 261708, 32400, 2292, 80, 1, 167522976, 61343667, 18829650, 4271652, 657198, 63690, 3555, 99, 1, 2178520080, 811147590, 265116720, 67358500, 12269312, 1506615, 117040, 5280, 120, 1

Offset: 1

N. J. A. Sloane, Jul 22 2015

See Mullin (1967) for precise definition.

The sequence 1, 8, 72, 720,... in the first column has the same values as in A260039.

			Triangle begins:
1,
8,1,
72,15,1,
720,190,24,1,
...

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]

Row sums are A260041.

(k+1)*T(n,k) = A260039(n,k), n>=1, 0<=k

Conjecture: T(n,n-2) = A005563(n) = 8, 15, 24,.... for n>=2. - R. J. Mathar, Jul 22 2015

Conjecture: T(n,n-3)= (n+1)*n*(5*n^2+7*n+6)/12 = 72, 190,.... for n>=3. - R. J. Mathar, Jul 22 2015

A260041 Number of twin-tree-rooted maps with n edges.

Original entry on oeis.org

0, 1, 9, 88, 935, 10576, 125398, 1541842, 19510569, 252692488, 3336041258, 44754332228, 608627917092, 8374308526624, 116400823021244, 1632397871933462, 23073139088034053, 328412887616247352, 4703787284447392654, 67751302345297146628, 980837632950784897364

Offset: 0

N. J. A. Sloane, Jul 22 2015

nonn

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]

Row sums of A260040.

Showing 1-5 of 5 results.

cp's OEIS Frontend

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

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A005568 Product of two successive Catalan numbers C(n)*C(n+1).

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A046715 Secondary root edges in doubly rooted tree maps with n edges.

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A260040 Triangle read by rows giving numbers H(n,k), number of classes of twin-tree-rooted maps with n edges whose root bond contains k edges.

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A260041 Number of twin-tree-rooted maps with n edges.

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