A234513 8*binomial(9*n+8,n)/(9*n+8).

1, 8, 100, 1496, 24682, 433160, 7932196, 149846840, 2898753715, 57135036024, 1143315429776, 23166186450680, 474347963242860, 9799792252101016, 204022381037886400, 4276098770070159096, 90151561242584838605, 1910564646571462338800

Offset

0,2

Comments

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), where p=9, r=8.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906v1, Discrete Math., 308 (2008), 4660-4669.

Thomas A. Dowling, Catalan Numbers Chapter 7.

Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.

Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.

Formula

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=9, r=8.

From Wolfdieter Lang, Feb 06 2020: (Start)

G.f.: hypergeom([8, 9, ..., 16]/9, [9, 10, ..., 16]/8, (9^9/8^8)*x).

E,g,f.: hypergeom([8, 10, 11, ..., 16]/9, [9, 10,..., 16]/8, (9^9/8^8)*x). Cf. _Ilya Gutkovsky_ in A118971. (End)

D-finite with recurrence 128*(8*n+3)*(4*n+3)*(8*n+1)*(2*n+1)*(8*n+7)*(4*n+1)*(8*n+5)*(n+1)*a(n) -81*(9*n+2)*(9*n+4)*(3*n+2)*(9*n-1)*(9*n+1)*(3*n+1)*(9*n+5)*(9*n+7)*a(n-1)=0. - R. J. Mathar, Aug 01 2022

From Wolfdieter Lang, Feb 15 2024: (Start)

a(n) = binomial(9*n+7, n+1)/(8*n+7), which is instance k = 8 of c(k, n+1) given in A130564.

The g.f. given above, and called B in the first line above, satisfies B(x)*(1 - x*B(x))^8 = 1. For the analog proof of the equivalence see A234466. x*B(x) is the compositional inverse of y*(1 - y)^8.

Another formula for the g.f. is B(x) = (8/(9*x))*(1 - 8F7([-1,1,2,3,4,5,6.7]/9, [1,2,3,4,5,6.7]/8; (9^9/8^8)*x)). (End)

Mathematica

Table[8 Binomial[9 n + 8, n]/(9 n + 8), {n, 0, 30}] (* Vincenzo Librandi, Dec 28 2013 *)

Prog

(PARI) a(n) = 8*binomial(9*n+8, n)/(9*n+8);
(PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(9/8))^8+x*O(x^n)); polcoeff(B, n)}
(Magma) [8*Binomial(9*n+8, n)/(9*n+8): n in [0..30]]; // Vincenzo Librandi, Dec 28 2013

Crossrefs

Cf. A000108, A118971, A130564, A143554, A234466, A234505, A234506, A234507, A234508, A234509, A234510, A232265.

Sequence in context: A261800 A208705 A246237 * A251686 A306032 A274844

Adjacent sequences: A234510 A234511 A234512 * A234514 A234515 A234516

Keyword

nonn,easy

Author

Tim Fulford, Dec 27 2013

Status

approved