A233830 - OEIS

%I #20 Sep 08 2022 08:46:06

%S 1,10,105,1170,13640,164502,2036265,25727800,330482295,4303216330,

%T 56672074888,753573733050,10103474312100,136435868978220,

%U 1854009194816745,25333847134998864,347880174736462550,4798137522234602700,66441427922465470095,923346006310186106010,12873823246049001482400

%N a(n) = 5*binomial(6*n+10,n)/(3*n+5).

%C Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=6, r=10.

%H Vincenzo Librandi, <a href="/A233830/b233830.txt">Table of n, a(n) for n = 0..200</a>

%H J-C. Aval, <a href="http://arxiv.org/pdf/0711.0906v1.pdf">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, here p=6, r=10.

%F From _Ilya Gutkovskiy_, Sep 14 2018: (Start)

%F E.g.f.: 6F6(5/3,11/6,2,13/6,7/3,5/2; 1,11/5,12/5,13/5,14/5,3; 46656*x/3125).

%F a(n) ~ 3^(6*n+19/2)*4^(3*n+5)/(sqrt(Pi)*5^(5*n+19/2)*n^(3/2)). (End)

%t Table[5 Binomial[6 n + 10, n]/(3 n + 5), {n, 0, 30}]

%o (PARI) a(n) = 5*binomial(6*n+10,n)/(3*n+5);

%o (PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(3/5))^10+x*O(x^n)); polcoeff(B, n)}

%o (Magma) [5*Binomial(6*n+10, n)/(3*n+5): n in [0..30]];

%Y Cf. A000108, A002295, A212071, A212072, A212073, A130564, A233743, A233827, A233829.

%K nonn

%O 0,2

%A _Tim Fulford_, Dec 16 2013