A093197 Number of labeled plane 2-trees on n triangles.

1, 6, 56, 720, 11880, 240240, 5765760, 160392960, 5079110400, 180503769600, 7117005772800, 308403583488000, 14572069319808000, 745747076954880000, 41098950018846720000, 2427001153744527360000, 152901072685905223680000

Offset: 2

Author

Ralf Stephan, Apr 22 2004

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 2..200
• G. Labelle, C. Lamathe and P. Leroux, A classification of plane and planar 2-trees, arXiv:math/0202052 [math.CO], 2002.

Programs

Mathematica

a[n_]:=n! * (2/3 * CatalanNumber[n] - 1/6 * CatalanNumber[n+1]); Array[a,50,2] (* Stefano Spezia, Sep 01 2018 *)

PARI

a(n)=if(n < 2, 0, n!*(4*binomial(2*n,n)/(n+1)-binomial(2*n+2,n+1)/(n+2))/6) \\ Andrew Howroyd, Aug 31 2018

Formula

a(n) = n! * (2/3 * C(n) - 1/6 * C(n+1)), where C(n) = A000108(n).

a(n) = RisingFactorial(5 + n, n) assuming offset 0. - Peter Luschny, Mar 22 2022