0,3

a(n), n>=1, enumerates sextic (6-ary) trees (rooted, ordered, incomplete) with n vertices (including the root).

Pfaff-Fuss-Catalan sequence C^{m}_n for m=6. See the Graham et al. reference, p. 347. eq. 7.66. See also the P\'olya-Szeg\"o reference.

Also 6-Raney sequence. See the Graham et al. reference, p. 346-7.

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.

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

Editor's note: "Ueber die Bestimmung der Anzahl der verschiedenen Arten, auf welche sich ein n-Eck durch Diagonalen in lauter m-Ecke zerlegen laesst, mit Bezug auf einige Abhandlungen der Herren Lame, Rodrigues, Binet, Catalan und Duhamel in dem Journal de Mathematiques pure et appliquees, publie par Joseph Liouville. T. III. IV.", Archiv der Mathematik u. Physik, 1 (1841), pp. 193ff; see especially p. 198.

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 288

O.g.f. A(x)= 1 + x*A(x)^6 = 1/(1-x*A(x)^5).

a(n)=binomial(6*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)=6 sextic trees (vertex degree <=6 and 6 possible branchings) with 2 vertices (one of them the root). Adding one more branch (one more vertex) to these 6 trees yields 6*6+binomial(6,2)=51=a(3) such trees.

Table[Binomial[6n, n]/(5n + 1), {n, 0, 20}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 06 2006

Cf. A002294, A002296.

Fifth column of triangle A062993.

Sequence in context: A133395 A050916 A011790 this_sequence A027393 A124565 A057817

Adjacent sequences: A002292 A002293 A002294 this_sequence A002296 A002297 A002298

easy,nonn,nice

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

More terms from Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 06 2006

Pfaff-Fuss-Catalan, Raney and 6-ary tree comments from W. Lang, Sep 14 2007.