A006631 From generalized Catalan numbers.

1, 8, 52, 320, 1938, 11704, 70840, 430560, 2629575, 16138848, 99522896, 616480384, 3834669566, 23944995480, 150055305008, 943448717120, 5949850262895, 37628321318280, 238591135349700, 1516500543586560, 9660632784642840, 61670325204822048, 394451619337629792

Offset: 0

Author

Simon Plouffe

Keywords

• nonn
• easy
• changed

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
• Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.

Crossrefs

Column 4 of A092276.

Cf. A006630.

Programs

Magma

A006631:= func< n | 4*Binomial(3*n+7,n)/(n+4) >;
[A006631(n): n in [0..40]]; // G. C. Greubel, Aug 31 2025

Mathematica

Table[SeriesCoefficient[HypergeometricPFQ[{3,8/3,10/3},{5,9/2},27*x/4],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 07 2012 *)
Table[4*Binomial[3*n+7,n]/(n+4), {n,0,40}] (* G. C. Greubel, Aug 31 2025 *)

PARI

a(n) = 8*binomial(3*n + 8, n)/(3*n + 8);

SageMath

def A006631(n): return 4*binomial(3*n+7,n)//(n+4)
print([A006631(n) for n in range(41)]) # G. C. Greubel, Aug 31 2025

Formula

G.f.: hypergeometric3_F_2([ 3, 8/3, 10/3 ], [ 5, 9/2 ], 27*x/4).

Recurrence: 2*(n+4)*(2*n+7)*a(n) = (5*n+13)*(11*n+29)*a(n-1) - 7*(31*n^2+87*n+62)*a(n-2) + 21*(3*n-1)*(3*n+1)*a(n-3). - Vaclav Kotesovec, Oct 07 2012

a(n) ~ 3^(3*n+15/2)/(2^(2n+6)*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 07 2012

a(n) = 8*binomial(3*n + 8, n)/(3*n + 8). - Andrew Howroyd, Nov 06 2017

Extensions

More terms from Vincenzo Librandi, May 03 2013