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URL: https://oeis.org/A230547

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A230547
a(n) = 3*binomial(3*n+9, n)/(n+3).
4
1, 9, 63, 408, 2565, 15939, 98670, 610740, 3786588, 23535820, 146710476, 917263152, 5752004349, 36174046743, 228124619100, 1442387942520, 9142452842985, 58083251802345, 369816259792035, 2359448984037600, 15082416490309740, 96586612269316884, 619586741695427928
OFFSET
0,2
COMMENTS
Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=3, r=9.
LINKS
Jean-Christophe Aval, Multivariate Fuss-Catalan Numbers, Discrete Math., Vol. 308, No. 20 (2008), 4660-4669; arXiv preprint, arXiv:0711.0906 [math.CO], 2007.
Thomas A. Dowling, Catalan Numbers, Chapter 7 of Applications of discrete mathematics, John G. Michaels and Kenneth H. Rosen (eds.), McGraw-Hill, New York, 1991. [Wayback Machine link]
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15 (2010), 939-955.
Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.
FORMULA
G.f. satisfies: A(x) = {1 + x*A(x)^(p/r)}^r, here p=3, r=9.
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
a(n) ~ 3^(3*n+21/2) / (4^(n+5) * n^(3/2) * sqrt(Pi)). - Amiram Eldar, Sep 12 2025
MATHEMATICA
Table[9 Binomial[3 n + 9, n]/(3 n + 9), {n, 0, 30}]
PROG
(PARI) a(n) = 9*binomial(3*n+9, n)/(3*n+9);
(PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(3/9))^9+x*O(x^n)); polcoeff(B, n)}
(Magma) [9*Binomial(3*n+9, n)/(3*n+9): n in [0..30]];
KEYWORD
nonn
AUTHOR
Tim Fulford, Oct 23 2013
STATUS
approved