A234573 a(n) = 9*binomial(10*n+9,n)/(10*n+9).

1, 9, 126, 2109, 38916, 763686, 15636192, 330237765, 7141879503, 157366449604, 3520256293710, 79735912636302, 1825080422272800, 42148579533938784, 980892581545169496, 22980848343194476245, 541581608172776494554, 12829884648994115426295, 305349921559399354716430

Offset

0,2

Comments

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), where p=10, r=9.

Links

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

J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906 [math.CO], 2007; Discrete Math., 308 (2008), 4660-4669.

Thomas A. Dowling, Catalan Numbers Chapter 7.

Elżbieta Liszewska and 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.

J. Sawada, J. Sears, A. Trautrim, and A. Williams, Demystifying our Grandparent's De Bruijn Sequences with Concatenation Trees, arXiv:2308.12405 [math.CO], 2023.

Formula

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=10, r=9.

From Wolfdieter Lang, Feb 06 2020: (Start)

G.f.: hypergeom([9, 10, ..., 18]/10, [10, 11, ..., 18]/9, (10^10/9^9)*x).

E.g.f.: hypergeom([9, 11, 12, ..., 18]/10, [10, 11, ..., 18]/9, (10^10/9^9) * x). Cf. _Ilya Gutkovsky_ in A118971. (End)

a(n) = binomial(10*n + 8 , n+1)/(9*n + 8) which is instance k = 9 of c(k, n+1) given in a comment in A130564. x*B(x), with the above given g.f. B(x), is the compositional inverse of y*(1 - y)^9, hence B(x)*(1 - x*B(x))^9 = 1. For another formula for B(x) involving the hypergeometric function 9F8 see the analog formula in A234513. - Wolfdieter Lang, Feb 15 2024

Mathematica

Table[9 Binomial[10 n + 9, n]/(10 n + 9), {n, 0, 30}]

Prog

(PARI) a(n) = 9*binomial(10*n+9, n)/(10*n+9);
(PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(10/9))^9+x*O(x^n)); polcoeff(B, n)}
(Magma) [9*Binomial(10*n+9, n)/(10*n+9): n in [0..30]];

Crossrefs

Cf. A000108, A059968, A118971, A130564, A234513, A234525, A234526, A234527, A234528, A234529, A234570, A234571, A229963.

Sequence in context: A261801 A065086 A246238 * A306033 A065707 A338077

Adjacent sequences: A234570 A234571 A234572 * A234574 A234575 A234576

Keyword

nonn,easy

Author

Tim Fulford, Dec 28 2013

Status

approved