0,3

The number of rooted loopless n-edge maps in the plane (planar with a distinguished outside face). - Valery A. Liskovets (liskov(AT)im.bas-net.by), 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

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 P\'olya-Szeg\"o reference.

Also 4-Raney sequence. See the Graham et al. reference, p. 346-7.

Bacher: "We describe the statistics of checkerboard triangulations obtained by coloring black every other triangle in triangulations of convex polygons." 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 fnctions. - Jonathan Vos Post (jvospost3(AT)gmail.com), Oct 05 2007

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

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

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).

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

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

G. P\'olya and G. Szeg\"o, Problems and Theorems in Analysis, Springer-Verlag, Heidelberg, New York, 2 vols., 1972, Vol. 1, problem 211, p. 146 with solution on p. 348.

T. D. Noe, Table of n, a(n) for n=0..100

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

Joerg Arndt, Fxtbook

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.

Roland Bacher, Fair Triangulations

O.g.f. 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.

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.

CoefficientList[InverseSeries[ Series[ y - y^4, {y, 0, 60}], x], x][[Range[2, 60, 3]]]

Cf. A001764, A002294, A000260, A027836.

Third column of triangle A062993.

Sequence in context: A142984 A097593 A025756 this_sequence A003287 A077056 A045744

Adjacent sequences: A002290 A002291 A002292 this_sequence A002294 A002295 A002296

nonn,nice

N. J. A. Sloane (njas(AT)research.att.com).

Pfaff-Fuss-Catalan, Raney and quartic tree comments from W. Lang, Sep 14 2007.