A233669 - OEIS

%I #29 Sep 08 2022 08:46:06

%S 1,7,56,490,4550,44051,439824,4496388,46834095,495260150,5303177880,

%T 57385471962,626548297648,6893781417320,76362138282400,

%U 850867975145160,9530515916642385,107249427630005661,1211964598880990640,13747501038498835300

%N a(n) = 7*binomial(5*n+7, n)/(5*n+7).

%C Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=5, r=7.

%H Vincenzo Librandi, <a href="/A233669/b233669.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=5, r=7.

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

%F E.g.f.: 4F4(7/5,8/5,9/5,11/5; 1,9/4,5/2,11/4; 3125*x/256).

%F a(n) ~ 7*5^(5*n+13/2)/(sqrt(Pi)*2^(8*n+31/2)*n^(3/2)). (End)

%t Table[7 Binomial[5 n + 7, n]/(5 n + 7), {n, 0, 30}]

%o (PARI) a(n) = 7*binomial(5*n+7,n)/(5*n+7);

%o (PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(5/7))^7+x*O(x^n)); polcoeff(B, n)}

%o (Magma) [7*Binomial(5*n+7,n)/(5*n+7): n in [0..30]];

%Y Cf. A000108, A002294, A118969, A143546, A118971, A233668, A233736, A233737, A233738.

%K nonn,easy

%O 0,2

%A _Tim Fulford_, Dec 14 2013