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A007556 Number of 8-ary trees with n vertices.
(Formerly M4565)
26
1, 1, 8, 92, 1240, 18278, 285384, 4638348, 77652024, 1329890705, 23190029720, 410333440536, 7349042994488, 132969010888280, 2426870706415800, 44627576949364104, 826044435409399800, 15378186970730687400 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Shifts left when convolved three times.

From Wolfdieter Lang, Sep 14 2007: (Start)

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 Pólya-Szegő reference.

Also 8-Raney sequence. See the Graham et al. reference, p. 346-7.

(End)

REFERENCES

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

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 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).

LINKS

Harvey P. Dale, Table of n, a(n) for n = 0..750

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

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 290

FORMULA

a(n) = binomial(8*n, n)/(7*n+1) = binomial(8*n+1, n)/(8*n+1) = A062993(n+6,6).

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

a(0) = 1; a(n) = Sum_{i1+i2+..i8=n-1} a(i1)*a(i2)*...*a(i8) for n>=1. - Robert FERREOL, Apr 01 2015

a(n) = binomial(8*n,n-1)/n for n>=1, a(0) = 1 (from the Lagrange series of the o.g.f. A(x) with its above given implicit equation).

From Karol A. Penson, Mar 26 2015: (Start)

In Maple notation,

e.g.f.: hypergeom([1/8, 1/4, 3/8, 1/2, 5/8, 3/4, 7/8], [2/7, 3/7, 4/7, 5/7, 6/7, 1, 8/7],(2^24/7^7)*z);

o.g.f.: hypergeom([1/8, 1/4, 3/8, 1/2, 5/8, 3/4, 7/8], [2/7, 3/7, 4/7, 5/7, 6/7, 8/7],(2^24/7^7)*z);

a(n) are special values of Jacobi polynomials, in Maple notation:

  a(n) = JacobiP(n-1, 7*n+1, -n, 1)/n, n=1,2,...

(End)

From Peter Bala, Oct 14 2015: (Start)

A(x)^2 is o.g.f. for A234461; A(x)^3 is o.g.f. for A234462;

A(x)^4 is o.g.f. for A234463; A(x)^5 is o.g.f. for A234464;

A(x)^6 is o.g.f. for A234465; A(x)^7 is o.g.f. for A234466;

A(x)^9 is o.g.f. for A234467. (End)

a(n) ~ 2^(24*n+1)/(sqrt(Pi)*7^(7*n+3/2)*n^(3/2)). - Ilya Gutkovskiy, Feb 07 2017

EXAMPLE

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.

MAPLE

seq(binomial(8*n+1, n)/(8*n+1), n=0..30); # Robert FERREOL, Apr 01 2015

n:=30: G:=series(RootOf(g = 1+x*g^8, g), x=0, n+1): seq(coeff(G, x, k), k=0..n); # Robert FERREOL, Apr 01 2015

MATHEMATICA

Table[Binomial[8n, n]/(7n+1), {n, 0, 20}] (* Harvey P. Dale, Dec 24 2012 *)

PROG

(Haskell)

a007556 0 = 1

a007556 n = a007318' (8 * n) (n - 1) `div` n

-- Reinhard Zumkeller, Jul 30 2013

(MAGMA) [Binomial(8*n, n)/(7*n+1): n in [0..20]]; // Vincenzo Librandi, Apr 02 2015

(PARI) vector(100, n, n--; binomial(8*n, n)/(7*n+1)) \\ Altug Alkan, Oct 14 2015

CROSSREFS

Seventh column of triangle A062993.

Cf. A007318, A234461, A234462, A234463, A234464, A234465, A234466, A234467.

Sequence in context: A266427 A239644 A099142 * A194042 A231618 A027395

Adjacent sequences:  A007553 A007554 A007555 * A007557 A007558 A007559

KEYWORD

nonn,nice,eigen

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified February 21 16:38 EST 2018. Contains 299414 sequences. (Running on oeis4.)