A052183 a(n) = (n + 2) * binomial(3*n, n) / (2*n + 1).

2, 3, 12, 60, 330, 1911, 11424, 69768, 432630, 2713425, 17168580, 109390320, 700939512, 4512458580, 29164264320, 189120846288, 1229917589262, 8018580361365, 52392620853300, 342991368096300, 2249282417749290

Offset

0,1

Comments

A second-order recursive sequence.

Links

Table of n, a(n) for n=0..20.

L. Carlitz, Enumeration of two-line arrays, Fib. Quart., 11 (1973), pp. 113-130.

Formula

a(n) = (n+2)*c(2; n), where c(2; n) = binomial(3*n, n)/(2*n+1) (A001764).

c(2; n) is equivalent to Eq. (6.22) on p. 129 of the Carlitz reference.

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

G.f.: (2-5*g)/((3*g-1)*(g-1)) where g*(1-g)^2 = x. - Mark van Hoeij, Nov 10 2011

Conjecture: 2*n*(2*n+1)*a(n) + (-47*n^2+50*n-12)*a(n-1) + 15*(3*n-4)*(3*n-5)*a(n-2) = 0. - R. J. Mathar, Sep 27 2012

Mathematica

Table[(n+2) Binomial[3n, n]/(2n+1), {n, 0, 20}] (* Harvey P. Dale, Mar 23 2013 *)

Crossrefs

Cf. A001764, A000108, A000139.

Sequence in context: A094532 A092980 A191464 * A123899 A188588 A032133

Adjacent sequences: A052180 A052181 A052182 * A052184 A052185 A052186

Keyword

easy,nonn

Author

Barry E. Williams, Jan 27 2000

Extensions

More terms from James A. Sellers, Jan 31 2000

New name using a formula of the author by Peter Luschny, Feb 23 2024

Status

approved