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

0,2

Comments

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

Links

Table of n, a(n) for n=0..30.

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

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

Example

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.

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 *)

Crossrefs

Column k=5 of A350248.

Cf. A350116.

Sequence in context: A028530 A200836 A276980 * A214220 A029567 A028534

Adjacent sequences: A350300 A350301 A350302 * A350304 A350305 A350306

Keyword

easy,nonn

Author

Janaka Rodrigo, Dec 24 2021

Status

approved