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A002293 Number of dissections of a polygon: binomial(4n,n)/(3n+1).
(Formerly M3587 N1454)
1, 1, 4, 22, 140, 969, 7084, 53820, 420732, 3362260, 27343888, 225568798, 1882933364, 15875338990, 134993766600, 1156393243320, 9969937491420, 86445222719724, 753310723010608, 6594154339031800, 57956002331347120, 511238042454541545 (list; graph; refs; listen; history; text; internal format)



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 (3n+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 = 1/3x (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. See also the Pólya-Szegő reference.

Also 4-Raney sequence. See the Graham et al. reference, pp. 346-347.


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

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], in Maple notation:

a(n) = int(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)), x=0..256/27), n=0,1,... .

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)

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 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' 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

From Noemie Combe Feb 28 2017:

This is the number of A'Campo bi-colored forests of degree n and co-dimension 0. This can be shown using generating functions or a combinatorial approach. See link below.


R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, pp. 200, 347.

V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.

G. Pólya and G. Szegő, Problems and Theorems in Analysis, Springer-Verlag, Heidelberg, New York, 2 vols., 1972, Vol. 1, problem 211, p. 146 with solution on p. 348.

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).

L. Takacs, Enumeration of rooted trees and forests, Math. Scientist 18 (1993), 1-10, esp. Eq. (5).


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]

Joerg Arndt, Matters Computational (The Fxtbook), pp. 337-338

J. Arndt, Subset-lex: did we miss an order?, arXiv:1405.6503 [math.CO], 2014-2015.

Roland Bacher, Fair Triangulations, arXiv:0710.0960 [math.CO], 2007.

C. Banderier, M. Bousquet-Mélou, A. Denise, P. Flajolet, D. Gardy and D. Gouyou-Beauchamps, Generating Functions for Generating Trees, Discrete Mathematics 246(1-3), March 2002, pp. 29-55.

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210.

D. Bevan, D. Levin, P. Nugent, J. Pantone, 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.

J. Cigler, Some remarks about q-Chebyshev polynomials and q-Catalan numbers and related results, 2013.

T. Copeland, Discriminating Deltas, Depressed Equations, and Generalized Catalan Numbers, 2012.

M. Gross, Mirror symmetry and the Strominger-Yau-Zaslow conjecture, arXiv preprint arXiv:1212.4220 [math.AG], p. 58, 2013.

F. Harary, E. M. Palmer and R. C. Read, On the cell-growth problem for arbitrary polygons, Discr. Math. 11 (1975), 371-389.

V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., 14 (1976), 395-405.

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.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 286

V. A. Liskovets and T. R. Walsh, Counting unrooted maps on the plane, Advances in Applied Math., 36 No. 4 (2006), 364-387.

R. P. Loh, A. G. Shannon, A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980.

D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344 (T_n for s=4).

J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.

C. B. Pah and M. Saburov, Single Polygon Counting on Cayley Tree of Order 4: Generalized Catalan Numbers, Middle-East Journal of Scientific Research 13 (Mathematical Applications in Engineering): 01-05, 2013, ISSN 1990-9233.

Karol A. Penson and Karol Zyczkowski, Product of Ginibre matrices : Fuss-Catalan and Raney distribution, Phys. Rev. E 83, 061118, 15 June 2011.

Karol A. Penson and Karol Zyczkowski, Product of Ginibre matrices : Fuss-Catalan and Raney distribution, arXiv:1103.3453 [math-ph], 2011.

Alison Schuetz and Gwyneth Whieldon, Polygonal Dissections and Reversions of Series, arXiv:1401.7194 [math.CO], 2014.

B. Sury, Generalized Catalan numbers: linear recursion and divisibility, JIS 12 (2009) 09.7.5

Wikipedia, Fuss-Catalan number

S. Yakoubov, Pattern Avoidance in Extensions of Comb-Like Posets, arXiv preprint arXiv:1310.2979 [math.CO], 2013-2014.

N.Combe, V.Jugé,Counting bi-colored A'Campo forets arxiv:1702.07672 [Math AG], 2017


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.

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

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, j=0..n-1-i, 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)

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


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.


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


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 *)


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

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

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


from __future__ import division

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


Cf. A001764, A002294, A000260, A027836.

Third column of triangle A062993.

Cf. A069271, A006632, A196678, A006633, A233658, A233666, A006634,  A233667.

A283049, A277877, A283102,A283101,A283103

Sequence in context: A187254 A216712 A240586 * A181784 A003287 A077056

Adjacent sequences:  A002290 A002291 A002292 * A002294 A002295 A002296




N. J. A. Sloane


I added a contribution, since the Fuss-Catalan number is also the number of bi -colored A'Campo Forets of co-dimension 0.



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Last modified March 23 20:15 EDT 2017. Contains 283957 sequences.