A350599 Number of ways to partition the set of vertices of a convex n-gon into non-intersecting directed polygons.

2, 2, 2, 14, 30, 50, 170, 462, 1014, 2810, 7906, 19910, 53278, 148514, 397530, 1073918, 2976390, 8172426, 22413266, 62219830, 172846382, 479683762, 1338281802, 3743620974, 10475828630, 29389158426, 82643684034, 232644515366, 655928162878, 1852640651330, 5239096953274

Offset: 3

Author

Janaka Rodrigo, Jan 08 2022

Keywords

• nonn

Comments

A directed polygon is a polygon with an associated direction (clockwise or counterclockwise).

Equivalently, the polygons can be colored using two colors. - Andrew Howroyd, Jan 09 2022

Examples

			a(7) = 2 + 28 = 30 since the 7-gon can be given two directions and the 7-gon can also be partitioned into a triangle and a quadrilateral in 7 different ways giving another 7 * 4 = 28 possibilities.
		

Crossrefs

Cf. A350116, A350248.

Programs

Mathematica

a[n_] := Sum[2^k * Binomial[n + 1, k] * Binomial[n - 2*k - 1, k - 1]/(n + 1), {k, 1, Floor[n/3]}]; Array[a, 30, 3] (* Amiram Eldar, Jan 08 2022 *)

PARI

a(n) = sum(k=1, n\3, 2^k * binomial(n+1, k) * binomial(n-2*k-1, k-1)) / (n+1) \\ Andrew Howroyd, Jan 08 2022

Formula

a(n) = Sum_{k=1..floor(n/3)} 2^k * binomial(n+1, k) * binomial(n-2*k-1, k-1) / (n+1).

a(n) = Sum_{k=1..floor(n/3)} 2^k * A350248(n,k). - Andrew Howroyd, Jan 09 2022

The compositional inverse of x+Sum_{k=1..infinity} a_k x^{k+1} is x(1-x)/(1+x)(1-2x+x^2). Proved at MathOverflow 418996.