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A230547 a(n) = 3*binomial(3*n+9, n)/(n+3).

%I A230547 #35 Nov 22 2024 07:43:59
%S A230547 1,9,63,408,2565,15939,98670,610740,3786588,23535820,146710476,
%T A230547 917263152,5752004349,36174046743,228124619100,1442387942520,
%U A230547 9142452842985,58083251802345,369816259792035,2359448984037600
%N A230547 a(n) = 3*binomial(3*n+9, n)/(n+3).
%C A230547 Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=3, r=9.
%H A230547 Vincenzo Librandi, <a href="/A230547/b230547.txt">Table of n, a(n) for n = 0..200</a>
%H A230547 J-C. Aval, <a href="https://arxiv.org/abs/0711.0906">Multivariate Fuss-Catalan Numbers</a>, arXiv:0711.0906 [math.CO], 2007; Discrete Math., 308 (2008), 4660-4669.
%H A230547 Thomas A. Dowling, <a href="http://www.mhhe.com/math/advmath/rosen/r5/instructor/applications/ch07.pdf">Catalan Numbers Chapter 7</a>
%H A230547 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.
%H A230547 Emanuele Munarini, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Munarini/muna4.html">Shifting Property for Riordan, Sheffer and Connection Constants Matrices</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.
%F A230547 G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, here p=3, r=9.
%F A230547 D-finite with recurrence 2*n*(2*n+9)*(n+4)*a(n) -3*(3*n+7)*(n+2)*(3*n+8)*a(n-1)=0. - _R. J. Mathar_, Nov 22 2024
%t A230547 Table[9 Binomial[3 n + 9, n]/(3 n + 9), {n, 0, 30}]
%o A230547 (PARI) a(n) = 9*binomial(3*n+9,n)/(3*n+9);
%o A230547 (PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(3/9))^9+x*O(x^n)); polcoeff(B, n)}
%o A230547 (Magma) [9*Binomial(3*n+9, n)/(3*n+9): n in [0..30]];
%Y A230547 Cf. A000108, A001764, A006013, A006629, A102893, A006630, A102594, A006631, A233657.
%K A230547 nonn
%O A230547 0,2
%A A230547 _Tim Fulford_, Oct 23 2013