login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A007556 Number of 8-ary trees with n vertices.
(Formerly M4565)
39
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)
This is instance k = 8 of the generalized Catalan family {C(k, n)}_{n>=0} given in a comment of A130564. - Wolfdieter Lang, Feb 05 2024
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).
LINKS
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:math/0205301 [math.CO], 2002]
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]
Clemens Heuberger, Sarah J. Selkirk, and Stephan Wagner, Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo k, arXiv:2204.14023 [math.CO], 2022.
Lajos Takács, Enumeration of rooted trees and forests, Math. Scientist 18 (1993), 1-10, esp. Eq. (5).
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
D-finite with recurrence: 7*n*(7*n-3)*(7*n+1)*(7*n-2)*(7*n-5)*(7*n-1)*(7*n-4)*a(n) -128*(8*n-5)*(4*n-1)*(8*n-7)*(2*n-1)*(8*n-1)*(4*n-3)*(8*n-3)*a(n-1)=0. - R. J. Mathar, Feb 20 2020
EXAMPLE
There are a(2) = 8 octic trees (vertex degree less than or equal to 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. A130564.
Sequence in context: A266427 A239644 A099142 * A194042 A231618 A346768
KEYWORD
nonn,nice,eigen
AUTHOR
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)