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

1, 5, 30, 200, 1425, 10626, 81900, 647280, 5217300, 42724825, 354465254, 2973052680, 25168220350, 214762810500, 1845308367000, 15951899986272, 138638564739180, 1210677947695620, 10617706139119000, 93477423115076000

Offset: 0

Author

Karol A. Penson, Oct 05 2011

Keywords

• nonn
• easy

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

References

• 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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..110
• C. B. 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.
• Karol A. Penson and Karol Zyczkowski, Product of Ginibre matrices : Fuss-Catalan and Raney distribution, arXiv version
• Wikipedia, Fuss-Catalan number
• Karol Zyczkowski, Karol A. Penson, Ion Nechita and Benoit Collins, Generating random density matrices, J. Math Phys. 52, 062201 (2011). arXiv version.

Crossrefs

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).

Programs

Magma

[5*Binomial(4*n+5,n)/(4*n+5): n in [0..30]]; // Vincenzo Librandi, Oct 07 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

Extensions

Offset changed from 1 to 0 and extended by Vincenzo Librandi, Oct 07 2011