A046646 a(n) = 2*binomial(3*n-3, n-1)/(2*n-1) for n >= 2, and a(1) = 1.

1, 2, 6, 24, 110, 546, 2856, 15504, 86526, 493350, 2861430, 16829280, 100134216, 601661144, 3645533040, 22249511328, 136657509918, 844061090670, 5239262085330, 32665844580600, 204480219795390, 1284624902435490

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Number of certain rooted planar maps.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• Daniel Birmajer, Juan B. Gil, Michael D. Weiner, Bounce statistics for rational lattice paths, arXiv:1707.09918 [math.CO], 2017, p. 10.
• W. G. Brown, Enumeration of non-separable planar maps, Canad. J. Math., 15 (1963), 526-545.
• W. G. Brown, Enumeration of non-separable planar maps [Annotated scanned copy]

Crossrefs

A diagonal of A046651.

Cf. A046647, A005470, A000139, A001764.

Programs

Magma

[1] cat [2*Binomial(3*n-3,n-1)/(2*n-1): n in [2..30]]; // Vincenzo Librandi, Oct 13 2013

Maple

alias(PS=ListTools:-PartialSums): A046646List := proc(m) local A, P, n;
A := [1,2]; P := [2]; for n from 1 to m - 2 do P := PS(PS([op(P), P[-1]]));
A := [op(A), P[-1]] od; A end: A046646List(22); # Peter Luschny, Mar 26 2022

Mathematica

Join[{1},Table[(2*Binomial[3n-3,n-1])/(2n-1),{n,2,30}]] (* Harvey P. Dale, Oct 12 2013 *)

Formula

From Emeric Deutsch, Mar 03 2004: (Start)

a(n) = 2*binomial(3*n-3, n-1)/(2*n-1) for n >= 2, and a(1) = 1.

a(n) = 2*A001764(n-1) for n >= 2. (End)

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

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

Extensions

More terms from Emeric Deutsch, Mar 03 2004

New name using a formula of Emeric Deutsch by Peter Luschny, Feb 23 2024