A352477 a(n) is the number of different ways to partition the set of vertices of a convex n-gon into 4 intersecting polygons.

15345, 199914, 1610700, 10333050, 57958005, 297787980, 1439757200, 6662364668, 29844152321, 130445708134, 559533869356, 2365296230374, 9885290683829, 40944327268760, 168389163026240, 688631375953560, 280357073972529, 11373212442818370, 46006062638648940

Offset: 12

Author

Janaka Rodrigo, Mar 17 2022

Keywords

• nonn

Examples

			The set of vertices of a convex 14-gon can be partitioned into 4 polygons in 1611610 different ways:
- 3 triangles and 1 pentagon in (1/3!)*C(14,3)*C(11,3)*C(8,3)*C(5,5) = 560560 different ways, and
- 2 triangles and 2 quadrilaterals in (1/2!)*(1/2!)*C(14,3)*C(11,3)*C(8,4)*C(4,4) = 1051050 ways.
Subtracting the A350286(14-11)=910 nonintersecting partitions leaves a(14)=1610700.
		

Crossrefs

Cf. A261724, A350116, A350286.

Programs

PARI

a4(n) = (1/12)*(-3^(n - 2)*(n^2 + 5*n + 18) + (1/64)*(2^(2*n + 5) + 3*2^n*(n^4 + 2*n^3 + 19*n^2 + 42*n + 64) - 16*(n^6 - 9*n^5 + 43*n^4 - 91*n^3 + 112*n^2 - 32*n + 8))); \\ A261724
a6(n) = (n*(n+1)*(n+2)*(n+9)*(n+10)*(n+11))/144; \\ A350286
a(n) = a4(n) - a6(n-11); \\ Michel Marcus, Mar 20 2022

Formula

a(n) = A261724(n) - A350286(n - 11), n > 11.