A062994 Eighth column of triangle A062993 (without leading zeros). A Pfaff-Fuss or 9-Raney sequence.

1, 1, 9, 117, 1785, 29799, 527085, 9706503, 184138713, 3573805950, 70625252863, 1416298046436, 28748759731965, 589546754316126, 12195537924351375, 254184908607118800, 5332692942907262361

Offset: 0

Author

Wolfdieter Lang, Jul 12 2001

Keywords

• nonn
• easy

Comments

See Graham et al., Hilton and Pedersen, Hoggat and Bicknell, Frey and Sellers references given in A062993.

Essentially the same as A059967. a(n), n>=1, enumerates 9-ary trees (rooted, ordered, incomplete) with n vertices (including the root).

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

Examples

			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.
		

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
• M. Gross, P. Hacking, S. Keel, and M. Kontsevich, Canonical bases for cluster algebras, arXiv preprint arXiv:1411.1394 [math.AG], 2016.

Crossrefs

Column k=8 of A070914.

Cf. A000108, A001764, A002293, A002294, A002295, A002296, A007556, A059968, A130564.

Programs

Maple

seq(binomial(9*k+1,k)/(8*k+1),k=0..30);
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

Table[Binomial[9n,n]/(8n+1),{n,0,30}] (* Harvey P. Dale, Oct 28 2012 *)

PARI

{ for (n=0, 100, write("b062994.txt", n, " ", binomial(9*n, n)/(8*n + 1)) ) } \\ Harry J. Smith, Aug 15 2009

Formula

a(n) = A062993(n+7, 7) = binomial(9*n, n)/(8*n+1).

G.f.: RootOf((_Z^9)*x-_Z+1) (Maple notation, from ECS, see links for A007556).

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

Extensions

9-ary tree comments and Pólya and G. Szegő reference from Wolfdieter Lang, Sep 14 2007