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a(n) = binomial(11*n+1,n)/(11*n+1).
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%I #57 Jul 19 2024 08:46:35

%S 1,1,11,176,3311,68211,1489488,33870540,793542167,19022318084,

%T 464333035881,11502251937176,288417894029200,7306488667126803,

%U 186719056586568660,4807757550367267056,124609430032127192295,3248403420844673986345

%N a(n) = binomial(11*n+1,n)/(11*n+1).

%C Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=11, r=1. Interesting property when r=1, a(n+1,p,1) = a(n,p,p) for n>=0.

%C This is also instance k = 10 of the generalized Catalan family {C(k, n)}_{n>=0} given in a comment of A130564. - _Wolfdieter Lang_, Feb 05 2024

%H Vincenzo Librandi, <a href="/A230388/b230388.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:0711.0906 [math.CO], 2008.

%H J-C. Aval, <a href="http://dx.doi.org/10.1016/j.disc.2007.08.100">Multivariate Fuss-Catalan Numbers</a>, 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.

%H J.-C. Novelli, J.-Y. Thibon, <a href="http://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions</a>, arXiv preprint arXiv:1403.5962, 2014

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Fuss-Catalan_numbers">Fuss-Catalan number</a>

%F G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, with p=11, r=1.

%F From _Robert FERREOL_, Apr 01 2015: (Start)

%F a(n) = binomial(11*n,n)/(10*n+1) = A062993(n+9, 9).

%F a(0) = 1; a(n) = Sum_{i1+i2+..i11=n-1} a(i1)*a(i2)*...*a(i11) for n>=1.

%F (End)

%F O.g.f.: hypergeometric([1,...,10]/11,[2,...,9,11]/10,(11^11/10^10)*x). For the e.g.f. put an extra 1 = 10/10 into the second part. - _Wolfdieter Lang_, Feb 05 2024

%F a(n) ~ (11^11/10^10)^n*sqrt(11/(2*Pi*(10*n)^3)). - _Robert A. Russell_, Jul 15 2024

%p seq(binomial(11*k+1,k)/(11*k+1),k=0..30); # _Robert FERREOL_, Apr 01 2015

%p n:=30:G:=series(RootOf(g = 1+x*g^11, g),x=0,n+1):seq(coeff(G,x,k),k=0..n); # _Robert FERREOL_, Apr 01 2015

%t Table[Binomial[11 n + 1, n]/(11 n + 1), {n, 0, 30}] (* _Vincenzo Librandi_, Jan 01 2014 *)

%o (PARI) a(n) = binomial(11*n+1,n)/(11*n+1);

%o (PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(11))+x*O(x^n)); polcoeff(B, n)}

%o (Magma) [Binomial(11*n+1,n)/(11*n+1): n in [0..30]]; // _Vincenzo Librandi_, Jan 01 2014

%Y Cf. A234868-A234873, A130564.

%K nonn,easy

%O 0,3

%A _Tim Fulford_, Jan 01 2014