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A196678 a(n) = 5*binomial(4*n+5,n)/(4*n+5).

%I A196678 #26 Sep 08 2022 08:45:59
%S A196678 1,5,30,200,1425,10626,81900,647280,5217300,42724825,354465254,
%T A196678 2973052680,25168220350,214762810500,1845308367000,15951899986272,
%U A196678 138638564739180,1210677947695620,10617706139119000,93477423115076000
%N A196678 a(n) = 5*binomial(4*n+5,n)/(4*n+5).
%C A196678 This is a sequence  of power moments of the following signed function defined on the segment (0,256/27), in Maple notation:
%C A196678 -(1/2)*sqrt(2)*x^(1/4)*hypergeom([-5/12, -1/12, 5/4], [1/2, 3/4], (27/256)*x)/Pi+(5/4)*sqrt(x)*hypergeom([-1/6, 1/6, 3/2], [3/4, 5/4], (27/256)*x)/Pi-(15/64)*sqrt(2)*x^(3/4)*hypergeom([1/12, 5/12, 7/4], [5/4, 3/2], (27/256)*x)/Pi. This function is not positive on (0,256/27).
%C A196678 The two parameter Fuss-Catalan sequence is A(n,p,r) := r*binomial(n*p + r, n)/(n*p + r). This sequence is A(n,4,5). - _Peter Bala_, Oct 16 2015
%D A196678 C. H. Pah, M. R. Wahiddin, Combinatorial Interpretation of Raney Numbers and Tree Enumerations, Open Journal of Discrete Mathematics, 2015, 5, 1-9; http://www.scirp.org/journal/ojdm; http://dx.doi.org/10.4236/ojdm.2015.51001
%H A196678 Vincenzo Librandi, <a href="/A196678/b196678.txt">Table of n, a(n) for n = 0..110</a>
%H A196678 C. B. Pah and M. Saburov, <a href="http://dx.doi.org/10.5829/idosi.mejsr.2013.13.mae.9991">Single Polygon Counting on Cayley Tree of Order 4: Generalized Catalan Numbers</a>, Middle-East Journal of Scientific Research 13 (Mathematical Applications in Engineering): 01-05, 2013, ISSN 1990-9233.
%H A196678 Karol A. Penson and Karol Zyczkowski, <a href="http://dx.doi.org/10.1103/PhysRevE.83.061118">Product of Ginibre matrices : Fuss-Catalan and Raney distribution</a>, <a href="http://arxiv.org/abs/1103.3453/">arXiv version</a>
%H A196678 Wikipedia, <a href="https://en.wikipedia.org/wiki/Fuss-Catalan_number">Fuss-Catalan number</a>
%H A196678 Karol Zyczkowski, Karol A. Penson, Ion Nechita and Benoit Collins, <a href="http://dx.doi.org/10.1063/1.3595693"> Generating random density matrices</a>, J. Math Phys. 52, 062201 (2011). <a href="http://arxiv.org/abs/1010.3570">arXiv version</a>.
%F A196678 O.g.f.: hypergeom([5/4, 3/2, 7/4], [7/3, 8/3], (256 z)/27)
%F A196678 E.g.f.: hypergeom([5/4, 3/2, 7/4], [1, 7/3, 8/3], (256 z)/27)
%F A196678 From _Peter Bala, Oct 16 2015: (Start)
%F A196678 O.g.f. A(x) = 1/x * series reversion (x*C(-x)^5), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. See cross-references for other Fuss-Catalan sequences with o.g.f. 1/x * series reversion (x*C(-x)^k), k = 3 through 11.
%F A196678 A(x)^(1/5) is the o.g.f. for A002293. (End)
%F A196678 D-finite with recurrence 3*n*(3*n+5)*(3*n+4)*a(n) -8*(4*n+1)*(2*n+1)*(4*n+3)*a(n-1)=0. - _R. J. Mathar_, Aug 01 2022
%o A196678 (Magma) [5*Binomial(4*n+5,n)/(4*n+5): n in [0..30]]; // _Vincenzo Librandi_, Oct 07 2011
%Y A196678 Cf. A000108, A002293, A000245 (k = 3), A006629 (k = 4), A233668 (k = 6), A233743 (k = 7), A233835 (k = 8), A234467 (k = 9), A232265 (k = 10), A229963 (k = 11).
%K A196678 nonn,easy
%O A196678 0,2
%A A196678 _Karol A. Penson_, Oct 05 2011
%E A196678 Offset changed from 1 to 0 and extended by _Vincenzo Librandi_, Oct 07 2011