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A350116 Number of ways to partition the set of vertices of a convex {n+8}-gon into 3 non-intersecting polygons.

%I A350116 #130 Aug 03 2022 07:14:26
%S A350116 0,12,45,110,220,390,637,980,1440,2040,2805,3762,4940,6370,8085,10120,
%T A350116 12512,15300,18525,22230,26460,31262,36685,42780,49600,57200,65637,
%U A350116 74970,85260,96570,108965,122512,137280,153340,170765,189630,210012,231990,255645,281060,308320
%N A350116 Number of ways to partition the set of vertices of a convex {n+8}-gon into 3 non-intersecting polygons.
%C A350116 Equivalently, the number of noncrossing set partitions of an {n+8}-set into 3 blocks with 3 or more elements in each block.
%H A350116 Andrew Howroyd, <a href="/A350116/b350116.txt">Table of n, a(n) for n = 0..1000</a>
%H A350116 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A350116 a(n) = n*(n+1)*(n+7)*(n+8)/12.
%F A350116 G.f.: -x*(12-15*x+5*x^2)/(x-1)^5 . - _R. J. Mathar_, Aug 03 2022
%e A350116 The a(1) = 12 solutions are:
%e A350116    {123}{456}{789}, {234}{567}{891}, {345}{678}{912},
%e A350116    {156}{234}{567}, {267}{345}{891}, {378}{456}{912},
%e A350116    {489}{567}{123}, {591}{678}{234}, {612}{789}{345},
%e A350116    {723}{891}{456}, {834}{912}{567}, {945}{123}{678}.
%e A350116 In the above, the numbers can be considered to be the partition of a 9-set into 3 blocks or the partition of the vertices of a convex 9-gon into 3 triangles (with the vertices labeled 1..9 in order).
%e A350116 a(2) = 45 corresponding to the number of ways to partition the vertices of a 10-gon into two triangles and one quadrilateral.
%t A350116 a[n_] := n*(n + 1)*(n + 7)*(n + 8)/12; Array[a, 40, 0] (* _Amiram Eldar_, Dec 21 2021 *)
%Y A350116 Column k=3 of A350248.
%Y A350116 The case of any number of parts for an n-gon is A114997.
%Y A350116 The case of exactly 2 parts for a {n+5}-gon is A055998.
%K A350116 easy,nonn
%O A350116 0,2
%A A350116 _Janaka Rodrigo_, Dec 21 2021