A006630 From generalized Catalan numbers. (Formerly M4214)

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

Offset

0,2

Comments

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

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

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.

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.: 3_F_2 ( [ 2, 8/3, 7/3 ]; [ 4, 7/2 ]; 27 x / 4 ).

a(n) = C(3n+6, n)*2/(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

Mathematica

Table[Binomial[3 n + 6, n] 2 / (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) [Binomial(3*n+6, n)*2/(n+2): n in [0..25]]; // Vincenzo Librandi, Aug 07 2015
(PARI) a(n) = binomial(3*n+6, n)*2/(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(binomial(3*n+6, n)*2/(n+2), n, 0, 30); /* Vladimir Kruchinin, Oct 03 2022 */

Crossrefs

Column 3 of A092276.

Closely related to A000139.

Sequence in context: A009162 A012718 A297221 * A367850 A180035 A360717

Adjacent sequences: A006627 A006628 A006629 * A006631 A006632 A006633

Keyword

nonn,easy

Author

Simon Plouffe

Extensions

More terms from Christopher Lund (clund(AT)san.rr.com), Apr 16 2002

a(21)-a(22) from Vincenzo Librandi, Aug 07 2015

Status

approved