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

%I A350303 #38 Feb 20 2022 20:29:02
%S A350303 0,273,1820,7140,21420,54264,122094,251370,482790,876645,1519518,
%T A350303 2532530,4081350,6388200,9746100,14535612,21244356,30489585,43044120,
%U A350303 59865960,82131896,111275472,149029650,197474550,259090650,336817845,434120778,555060870,704375490,887564720,1110986184
%N A350303 a(n) is the number of ways to partition the set of vertices of a convex (n+14)-gon into 5 nonintersecting polygons.
%C A350303 Equivalently, the number of noncrossing set partitions of an (n+14)-set into 5 blocks with 3 or more elements in each block.
%H A350303 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).
%F A350303 a(n) = (1/2880)*n*(n+1)*(n+2)*(n+3)*(n+11)*(n+12)*(n+13)*(n+14).
%F A350303 G.f.: 7*x*(39 - 91*x + 84*x^2 - 36*x^3 + 6*x^4)/(1 - x)^9. - _Stefano Spezia_, Dec 26 2021
%e A350303 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:
%e A350303   {1,2,3} {4,5,15}  {6,7,8}   {9,10,11} {12,13,14}
%e A350303   {1,2,3} {4,14,15} {5,6,7}   {8,9,10}  {11,12 13}
%e A350303   {1,2,3} {4,5,6}   {7,8,15}  {9,10,11} {12 13,14}
%e A350303   {1,2,3} {4,5,15}  {6,7,14}  {8,9,10}  {11,12,13}
%e A350303   {1,2,3} {4,14,15} {5,12,13} {6,7,8}   {9,10,11}
%e A350303   {1,2,3} {4,5,15}  {6,13,14} {7,8,9}   {10,11,12}
%e A350303   {1,2,3} {4,14,15} {5,6,13}  {7,8,9}   {10,11,12}
%e A350303   {1,2,3} {4,5,15}  {6,7,14}  {8,9,13}  {10,11,12}
%e A350303   {1,2,3} {4,5,15}  {6,7,14}  {8,12,13} {9,10,11}
%e A350303   {1,2,3} {4,5,15}  {6,13,14} {7,8,12}  {9,10,11}
%e A350303   {1,2,3} {4,14,15} {5,12,13} {6,7,11}  {8,9,10}
%e A350303   {1,2,3} {4,15,8}  {5,6,7}   {9,10,11} {12,13,14}
%e A350303   {1,2,3} {4,15,8}  {5,6,7}   {9,13,14} {10,11,12}
%e A350303   {1,2,3} {4,15,8}  {5,6,7}   {9,10,14} {11,12,13}
%e A350303   {1,2,3} {4,5,15}  {6,7,8}   {9,10,14} {11,12,13}
%e A350303   {1,2,3} {4,14,15} {5,6,7}   {8,12,13} {9,10,11}
%e A350303   {1,2,3} {4,14,15} {5,6,7}   {8,9,13}  {10,11,12}
%e A350303   {1,2,3} {4,5,15}  {6,7,8}   {9,13,14} {10,11,12}
%e A350303 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.
%e A350303 a(2)=1820 corresponding to the number of ways to partition the vertices of a 16-gon into 4 triangles and one quadrilateral.
%t A350303 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 *)
%Y A350303 Column k=5 of A350248.
%Y A350303 Cf. A350116.
%K A350303 easy,nonn
%O A350303 0,2
%A A350303 _Janaka Rodrigo_, Dec 24 2021