Voozh

A006630

From generalized Catalan numbers.

1, 6, 33, 182, 1020, 5814, 33649, 197340, 1170585, 7012200, 42364476, 257854776, 1579730984, 9734161206, 60290077905, 375138262520, 2343880406595, 14699630061270, 92502956574105, 583920410197950, 3696470074992240, 23461536762704040, 149270218961671548

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

It appears that this is the self-convolution of A001764 starting 1, 3, 12, ... . - Alon Regev, Aug 07 2015

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..200

Alin Bostan, Frédéric Chyzak, and Vincent Pilaud, Refined product formulas for Tamari intervals, arXiv:2303.10986 [math.CO], 2023-2024.

Paul Drube, Raised k-Dyck paths, arXiv:2206.01194 [math.CO], 2022. See Appendix pp. 14-15.

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.

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

FORMULA

G.f.: hypergeometric3_F_2([ 2, 8/3, 7/3 ], [ 4, 7/2 ], 27*x/4).

a(n) = 2*binomial(3*n+6, n)/(n+2). - Henry Bottomley, Sep 24 2001

G.f.: (1 - RootOf(x-t*(1-t)^2,t))^(-6) (algebraic function in Maple notation). - Mark van Hoeij, Nov 08 2011

G.f.: ((1/sqrt((3/4)*x)*sin((1/3)*asin(sqrt((27/4)*x)))-1)/x)^2. - Vladimir Kruchinin, Oct 03 2022

a(n) = (n+1)/2 * A000139(n+2). - F. Chapoton, Feb 23 2024

a(n) ~ 3^(3*n+13/2) / (4^(n+3) * n^(3/2) * sqrt(Pi)). - Amiram Eldar, Sep 12 2025

MATHEMATICA

Table[2 Binomial[3 n+6, n]/(n+2), {n, 0, 25}] (* Vincenzo Librandi, Aug 07 2015 *)

CoefficientList[Series[(-1 + (2*Sin[(1/3)*ArcSin[(3*Sqrt[3]*Sqrt[x])/2]]) / (Sqrt[3]*Sqrt[x]))^2/x^2, {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 03 2022, after Vladimir Kruchinin *)

PROG

(Magma) [2*Binomial(3*n+6, n)/(n+2): n in [0..25]]; // Vincenzo Librandi, Aug 07 2015

(PARI) a(n) = 2*binomial(3*n+6, n)/(n+2); \\ Andrew Howroyd, Nov 06 2017

(Maxima) taylor(((1/sqrt(3/4*x)*sin(1/3*asin(sqrt(27/4*x)))-1)/x)^2, x, 0, 17); /* Vladimir Kruchinin, Oct 03 2022 */

(Maxima) makelist(2*binomial(3*n+6, n)/(n+2), n, 0, 30); /* Vladimir Kruchinin, Oct 03 2022 */

(SageMath)

def A006630(n): return 2*binomial(3*(n+2), n)//(n+2)

print([A006630(n) for n in range(41)]) # G. C. Greubel, Aug 31 2025

CROSSREFS

Column 3 of A092276.

Closely related to A000139.

Cf. A001764, A006631.