A352900 a(n) is the number of different ways to partition the set of vertices of a convex n-gon into intersecting polygons.

0, 0, 0, 7, 28, 79, 460, 2486, 11209, 59787, 361777, 2167635, 13577211, 91919186, 650059294, 4761980740, 36508824672, 292116858616, 2424047807182, 20847409357919, 185754041370693, 1711253802075941, 16272637412753211, 159561718074359537, 1611599794862761838, 16747401536440092104

Offset: 3

Author

Janaka Rodrigo, Apr 07 2022

Keywords

• nonn

Examples

			For n=6, there are a(6) = 7 intersecting partitions of the convex hexagon. On vertices 1..6, they are the following pairs of triangles:
  {1,3,4}, {5,6,2}
  {4,5,1}, {2,3,6}
  {3,4,6}, {1,2,5}
  {2,3,5}, {1,4,6}
  {1,2,4}, {5,6,3}
  {1,6,3}, {5,4,2}
  {1,3,5}, {2,4,6}
		

Crossrefs

Cf. A006505, A059022, A114997, A350248.

Programs

PARI

T2(n,k) = if (n<3, 0, if (k==1, 1, k*T2(n-1,k) + binomial(n-1,2)*T2(n-3,k-1))); \\ A059022
a5(n) = if (n<3, n==0, sum(k=1, n\3, T2(n,k))); \\ A006505
a7(n) = sum(k=ceil((n+3)/2), n, (1/(n+1) * binomial(n+1, k) * binomial(2*k-n-3, n-k)) ); \\ A114997
a(n) =  a5(n) - a7(n); \\ Michel Marcus, Apr 09 2022

Formula

a(n) = A006505(n) - A114997(n).

a(n) = Sum_{k=2..floor(n/3)} (T(n,k) - C(n+1,k)*C(n-2k-1,k-1)/(n+1)); n > 5, where T(n,k) = k*T(n-1,k) + C(n-1,2)*T(n-3,k-1); n > 5 and 1 < k <= floor(n/3), T(n,k) = 1 when k = 1.

T(n,k) = A059022(n,k) is the number of different ways to partition the set of vertices of a convex n-gon into k polygons.

Extensions

More terms from Michel Marcus, Apr 09 2022