0,3

Shifts left when convolved thrice.

a(n), n>=1, enumerates octic (8-ary) trees (rooted, ordered, incomplete) with n vertices (including the root).

Pfaff-Fuss-Catalan sequence C^{m}_n for m=8. See the Graham et al. reference, p. 347. eq. 7.66. See also the P\'olya-Szeg\"o reference.

Also 8-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).

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.

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 290

C(8n, n)/(7n+1).

O.g.f. A(x)= 1 + x*A(x)^8 = 1/(1-x*A(x)^7).

a(n)=binomial(8*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)=8 octic trees (vertex degree <=8 and 8 possible branchings) with 2 vertices (one of them the root). Adding one more branch (one more vertex) to these 8 trees yields 8*8+binomial(8,2)=92=a(3) such trees.

Seventh column of triangle A062993.

Sequence in context: A155615 A133271 A099142 this_sequence A027395 A113353 A081624

Adjacent sequences: A007553 A007554 A007555 this_sequence A007557 A007558 A007559

nonn,nice,eigen

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

Pfaff-Fuss-Catalan, Raney, o.g.f. and 8-ary tree comments from W. Lang, Sep 14 2007.