A006634 From generalized Catalan numbers.

1, 9, 72, 570, 4554, 36855, 302064, 2504304, 20974005, 177232627, 1509395976, 12943656180, 111676661460, 968786892675, 8445123522144, 73940567860896, 649942898236596, 5733561315124260, 50744886833898400, 450461491952952690, 4009721145437152530, 35782256673785401065

Offset: 0

Author

Simon Plouffe

Keywords

• nonn

References

• H. M. Finucan, Some decompositions of generalized Catalan numbers, pp. 275-293 of Combinatorial Mathematics IX. Proc. Ninth Australian Conference (Brisbane, August 1981). Ed. E. J. Billington, S. Oates-Williams and A. P. Street. Lecture Notes Math., 952. Springer-Verlag, 1982.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Plouffe, Simon, Approximations of generating functions and a few conjectures, arXiv:0911.4975 [math.NT], 2002.

Programs

Maple

series(RootOf(g = 1+x*g^4,g)^9, x=0, 30); # Mark van Hoeij, Apr 22 2013

Mathematica

f[x_] := HypergeometricPFQ[ {9/4, 5/2, 11/4, 3}, {10/3, 11/3, 4}, 256/27*x]; Series[f[x], {x, 0, 16}] // CoefficientList[#, x]& (* Jean-François Alcover, Apr 23 2013, after Simon Plouffe *)

PARI

N = 3*66;  x = 'x + O('x^N);
g=serreverse(x-x^4)/x;
gf=g^9;  v=Vec(gf);
vector(#v\3,n,v[3*n-2])
/* Joerg Arndt, Apr 23 2013 */

Formula

G.f.: 4F3([9/4, 5/2, 11/4, 3],[10/3, 11/3, 4],256/27*x). - Simon Plouffe, Master's Thesis, UQAM, 1992

G.f.: g^9 where g = 1+x*g^4 is the g.f. of A002293. - Mark van Hoeij, Apr 22 2013

Extensions

More terms from Joerg Arndt, Apr 23 2013