OFFSET
0,3
COMMENTS
Eigth column of triangle A062993 (without leading zeros). A Pfaff-Fuss or 9-Raney sequence. - Wolfdieter Lang, Jul 12 2001
These numbers appear in a formula on p. 24 of Gross et al. for b = -2 or 4. For b = -1 or 3, see A002293. - Tom Copeland, Dec 24 2019
This is instance k=9 of the generalized Catalan family {C(k,n)}_{n>=0} given in a comment of A130564. - Wolfdieter Lang, Feb 05 2024
REFERENCES
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
LINKS
Harry J. Smith, Table of n, a(n) for n = 0..100
Mark Gross, Paul Hacking, Sean Keel, and Maxim Kontsevich, Canonical bases for cluster algebras, arXiv preprint arXiv:1411.1394 [math.AG], 2016.
Silvia Heubach, Nelson Y. Li, and Toufik Mansour, Staircase tilings and k-Catalan structures, Discrete Math., 308 (2008), 5954-5964.
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations, (m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014-2020.
FORMULA
G.f. A(x) satisfies: A = x + A^9.
a(n) = C(k*n, n)/((k-1)*n+1) = A062993(n+7, 7) = binomial(9*n, n)/(8*n+1), k=9.
G.f.: RootOf((_Z^9)*x-_Z+1) (Maple notation, from ECS, see links for A007556). - Wolfdieter Lang, Jul 12 2001
Recurrence: a(0) = 1; a(n) = Sum_{i1+i2+..+i9=n-1} a(i1)*a(i2)*...*a(i9) for n>=1. - Robert FERREOL, Apr 01 2015
From Ilya Gutkovskiy, Jan 16 2017: (Start)
O.g.f.: 8F7(1/9,2/9,1/3,4/9,5/9,2/3,7/9,8/9; 1/4,3/8,1/2,5/8,3/4,7/8,9/8; 387420489*x/16777216).
E.g.f.: 8F8(1/9,2/9,1/3,4/9,5/9,2/3,7/9,8/9; 1/4,3/8,1/2,5/8,3/4,7/8,1,9/8; 387420489*x/16777216).
a(n) ~ 3^(18*n+1)/(sqrt(Pi)*2^(24*n+5)*n^(3/2)). (End)
D-finite with recurrence: 128*n*(8*n-5)*(4*n-1)*(8*n+1)*(2*n-1)*(8*n-1)*(4*n-3)*(8*n-3)*a(n) -81*(9*n-7)*(9*n-5)*(3*n-1)*(9*n-1)*(9*n-8)*(3*n-2)*(9*n-4)*(9*n-2)*a(n-1)=0. - R. J. Mathar, Feb 20 2020
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^17). - Seiichi Manyama, Jun 16 2025
a(n) ~ 3^(18*n+1) / (2^(24*n+5) * n^(3/2) * sqrt(Pi)). - Amiram Eldar, Sep 29 2025
EXAMPLE
There are a(2)=9 9-ary trees (vertex degree <=9 and 9 possible branchings) with 2 vertices (one of them the root). Adding one more branch (one more vertex) to these 9 trees yields 9*9 + binomial(9,2) = 117 = a(3) such trees.
MAPLE
seq(binomial(9*k+1, k)/(8*k+1), k=0..30);
# Alternative:
n:=30: G:=series(RootOf(g = 1+x*g^9, g), x=0, n+1): seq(coeff(G, x, k), k=0..n); # Robert FERREOL, Apr 01 2015
MATHEMATICA
a[n_] := Binomial[9*n, n]/(8*n + 1); Array[a, 20, 0] (* Amiram Eldar, Sep 29 2025 *)
PROG
(PARI) a(n) = binomial(9*n, n)/(8*n + 1) \\ Harry J. Smith, Aug 15 2009
CROSSREFS
Related algebraic sequences concerning trees: strictly k-ary trees (A000108: s=x+s^2, A001263: s=(x, y)+(x, s)+(s, y)+(s, s))), (A001764: s=x+s^3), (A002293: s=x+s^4), (A002294: s=x+s^5), (A002295: s=x+s^6), (A002296: s=x+s^7), (A007556: s=x+s^8), at most k-ary trees (A001006: s=x+xs+xs^2), (A036765-A036769, s=x+xs^2....+xs^k, k=3, 4, 5, 6, 7).
Column k=8 of A070914.
KEYWORD
nonn,easy
AUTHOR
Claude Lenormand (claude.lenormand(AT)free.fr), Mar 05 2001
EXTENSIONS
a(0) = 1 inserted by Amiram Eldar, Sep 29 2025
STATUS
approved