0,3

a(n), n>=1, enumerates quintic trees (rooted, ordered, incomplete) with n vertices (including the root).

Pfaff-Fuss-Catalan sequence C^{m}_n for m=5. See the Graham et al. reference, p. 347. eq. 7.66. See also the P\'olya-Szeg\"o reference.

Also 5-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

Joerg Arndt, Fxtbook

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 287

O.g.f. A(x)= 1 + x*A(x)^5 = 1/(1-x*A(x)^4).

a(n)=binomial(5*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)=5 quintic trees (vertex degree <=5 and 5 possible branchings) with 2 vertices (one of them the root). Adding one more branch (one more vertex) to these five trees yields 5*5+binomial(5,2)=35=a(3) such trees.

CoefficientList[InverseSeries[ Series[ y - y^5, {y, 0, 100}], x], x][[Range[2, 100, 4]]]

Cf. A002295, A002296, A001764, A002293.

Fourth column of triangle A062993.

Sequence in context: A087630 A084135 A138233 this_sequence A051406 A000356 A027392

Adjacent sequences: A002291 A002292 A002293 this_sequence A002295 A002296 A002297

easy,nonn,nice

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

More terms from Olivier Gerard (olivier.gerard(AT)gmail.com), Jul 05 2001

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