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3*binomial(10*n+3,n)/(10*n+3).
9

%I #19 Sep 08 2022 08:46:06

%S 1,3,33,496,8610,162435,3235501,66959532,1425658806,31026962395,

%T 687124547340,15434728080408,350818684083868,8053515040969200,

%U 186457795206547635,4348790005989493960,102080931442008205230,2409777235191897422982

%N 3*binomial(10*n+3,n)/(10*n+3).

%C Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), where p=10, r=3.

%H Vincenzo Librandi, <a href="/A234526/b234526.txt">Table of n, a(n) for n = 0..200</a>

%H J-C. Aval, <a href="http://arxiv.org/abs/0711.0906">Multivariate Fuss-Catalan Numbers</a>, arXiv:0711.0906v1, Discrete Math., 308 (2008), 4660-4669.

%H Thomas A. Dowling, <a href="http://www.mhhe.com/math/advmath/rosen/r5/instructor/applications/ch07.pdf">Catalan Numbers Chapter 7</a>

%H Wojciech Mlotkowski, <a href="http://www.math.uiuc.edu/documenta/vol-15/28.pdf">Fuss-Catalan Numbers in Noncommutative Probability</a>, Docum. Mathm. 15: 939-955.

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

%t Table[3 Binomial[10 n + 3, n]/(10 n + 3), {n, 0, 30}] (* _Vincenzo Librandi_, Dec 28 2013 *)

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

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

%o (Magma) [3*Binomial(10*n+3, n)/(10*n+3): n in [0..30]]; // _Vincenzo Librandi_, Dec 28 2013

%Y Cf. A000108, A059968, A234525, A234527, A234528, A234529, A234570, A234571, A234573, A229963.

%K nonn

%O 0,2

%A _Tim Fulford_, Dec 27 2013