OFFSET
0,2
COMMENTS
This is a sequence of power moments of the following signed function defined on the segment (0,256/27), in Maple notation:
-(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).
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
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..110
Paul Drube, Raised k-Dyck paths, arXiv:2206.01194 [math.CO], 2022. See Appendix pp. 14-15.
C. H. Pah and M. Saburov, Single Polygon Counting on Cayley Tree of Order 4: Generalized Catalan Numbers, Middle-East Journal of Scientific Research 13 (Mathematical Applications in Engineering): 01-05, 2013, ISSN 1990-9233.
Chin Hee Pah and Mohamed Ridza Wahiddin, Combinatorial interpretation of raney numbers and tree enumerations, Open Journal of Discrete Mathematics, Vol. 5, No. 1 (2015), 1-9.
Karol A. Penson and Karol Życzkowski, Product of Ginibre matrices: Fuss-Catalan and Raney distributions, Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, Vol. 83, No. 6 (2011), 061118; arXiv preprint, arXiv:1103.3453 [math-ph], 2011.
Wikipedia, Fuss-Catalan number.
Karol Życzkowski, Karol A. Penson, Ion Nechita and Benoît Collins, Generating random density matrices, J. Math Phys. 52 (2011), 062201; arXiv preprint, arXiv:1010.3570 [quant-ph], 2010-2011.
FORMULA
O.g.f.: hypergeom([5/4, 3/2, 7/4], [7/3, 8/3], (256 z)/27)
E.g.f.: hypergeom([5/4, 3/2, 7/4], [1, 7/3, 8/3], (256 z)/27)
From Peter Bala, Oct 16 2015: (Start)
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.
A(x)^(1/5) is the o.g.f. for A002293. (End)
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
a(n) ~ 5 * 2^(8*n+17/2) / (3^(3*n+11/2) * n^(3/2) * sqrt(Pi)). - Amiram Eldar, Sep 12 2025
a(n-1) = n * A197271(n) / 2. - Robert A. Russell, Oct 10 2025
MATHEMATICA
a[n_] := 5*Binomial[4*n+5, n]/(4*n+5); Array[a, 20, 0] (* Amiram Eldar, Sep 12 2025 *)
PROG
(Magma) [5*Binomial(4*n+5, n)/(4*n+5): n in [0..30]]; // Vincenzo Librandi, Oct 07 2011
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Karol A. Penson, Oct 05 2011
EXTENSIONS
Offset changed from 1 to 0 and extended by Vincenzo Librandi, Oct 07 2011
STATUS
approved