A233657 a(n) = 10 * binomial(3*n+10,n)/(3*n+10).

1, 10, 75, 510, 3325, 21252, 134550, 848250, 5340060, 33622600, 211915132, 1337675430, 8458829925, 53591180360, 340185835500, 2163581913780, 13786238414025, 88004926973250, 562763873596575, 3604713725613000, 23126371951808268, 148594788106641360

Offset: 0

Author

Tim Fulford, Dec 14 2013

Keywords

• nonn
• easy

Comments

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=3, r=10.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• David Bevan, Robert Brignall, Andrew Elvey Price and Jay Pantone, A structural characterisation of Av(1324) and new bounds on its growth rate, arXiv preprint arXiv:1711.10325 [math.CO], 2017-2019.
• 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
• Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955 (2010).
• Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.

Crossrefs

Column 5 of A092276.

Cf. A000108, A001764, A006013, A006629, A102893, A006630, A102594, A006631, A230547.

Programs

Magma

[10*Binomial(3*n+10, n)/(3*n+10): n in [0..30]];

Maple

A233657:=n->10*binomial(3*n+10,n)/(3*n+10): seq(A233657(n), n=0..20); # Wesley Ivan Hurt, Oct 10 2014

Mathematica

Table[10 Binomial[3 n + 10, n]/(3 n + 10), {n, 0, 30}]

PARI

a(n) = 10*binomial(3*n+10,n)/(3*n+10);

PARI

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(3/10))^10+x*O(x^n)); polcoeff(B, n)}

Formula

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, here p=3, r=10.

+2*n*(n+5)*(2*n+9)*a(n) -3*(3*n+7)*(n+3)*(3*n+8)*a(n-1)=0. - R. J. Mathar, Feb 16 2018

E.g.f.: F([10/3, 11/3, 4], [1, 11/2, 6], 27*x/4), where F is the generalized hypergeometric function. - Stefano Spezia, Oct 08 2019