A350303 a(n) is the number of ways to partition the set of vertices of a convex (n+14)-gon into 5 nonintersecting polygons.

0, 273, 1820, 7140, 21420, 54264, 122094, 251370, 482790, 876645, 1519518, 2532530, 4081350, 6388200, 9746100, 14535612, 21244356, 30489585, 43044120, 59865960, 82131896, 111275472, 149029650, 197474550, 259090650, 336817845, 434120778, 555060870, 704375490, 887564720, 1110986184

Offset: 0

Author

Janaka Rodrigo, Dec 24 2021

Keywords

• easy
• nonn

Comments

Equivalently, the number of noncrossing set partitions of an (n+14)-set into 5 blocks with 3 or more elements in each block.

Examples

			The a(1)=273 solutions are {1,2,3} {4,5,6} {7,8,9} {10,11,12} {13,14,15} with its 3 different orientations and each of the following 18 patterns with its 15 orientations:
  {1,2,3} {4,5,15}  {6,7,8}   {9,10,11} {12,13,14}
  {1,2,3} {4,14,15} {5,6,7}   {8,9,10}  {11,12 13}
  {1,2,3} {4,5,6}   {7,8,15}  {9,10,11} {12 13,14}
  {1,2,3} {4,5,15}  {6,7,14}  {8,9,10}  {11,12,13}
  {1,2,3} {4,14,15} {5,12,13} {6,7,8}   {9,10,11}
  {1,2,3} {4,5,15}  {6,13,14} {7,8,9}   {10,11,12}
  {1,2,3} {4,14,15} {5,6,13}  {7,8,9}   {10,11,12}
  {1,2,3} {4,5,15}  {6,7,14}  {8,9,13}  {10,11,12}
  {1,2,3} {4,5,15}  {6,7,14}  {8,12,13} {9,10,11}
  {1,2,3} {4,5,15}  {6,13,14} {7,8,12}  {9,10,11}
  {1,2,3} {4,14,15} {5,12,13} {6,7,11}  {8,9,10}
  {1,2,3} {4,15,8}  {5,6,7}   {9,10,11} {12,13,14}
  {1,2,3} {4,15,8}  {5,6,7}   {9,13,14} {10,11,12}
  {1,2,3} {4,15,8}  {5,6,7}   {9,10,14} {11,12,13}
  {1,2,3} {4,5,15}  {6,7,8}   {9,10,14} {11,12,13}
  {1,2,3} {4,14,15} {5,6,7}   {8,12,13} {9,10,11}
  {1,2,3} {4,14,15} {5,6,7}   {8,9,13}  {10,11,12}
  {1,2,3} {4,5,15}  {6,7,8}   {9,13,14} {10,11,12}
In the above, the numbers can be considered to be the partition of a 15-set into 5 blocks or the partition of the vertices of a convex 15-gon into 5 triangles with vertices labeled 1,2,...,15 in order.
a(2)=1820 corresponding to the number of ways to partition the vertices of a 16-gon into 4 triangles and one quadrilateral.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Crossrefs

Column k=5 of A350248.

Cf. A350116.

Programs

Mathematica

a[n_] := n*(n + 1)*(n + 2)*(n + 3)*(n + 11)*(n + 12)*(n + 13)*(n + 14)/2880; Array[a, 30, 0] (* Amiram Eldar, Dec 26 2021 *)

Formula

a(n) = (1/2880)*n*(n+1)*(n+2)*(n+3)*(n+11)*(n+12)*(n+13)*(n+14).

G.f.: 7*x*(39 - 91*x + 84*x^2 - 36*x^3 + 6*x^4)/(1 - x)^9. - Stefano Spezia, Dec 26 2021