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A350286 Number of different ways to partition the set of vertices of a convex (n+11)-gon into 4 nonintersecting polygons.

%I A350286 #39 Dec 28 2021 04:25:23
%S A350286 0,55,286,910,2275,4900,9520,17136,29070,47025,73150,110110,161161,
%T A350286 230230,322000,442000,596700,793611,1041390,1349950,1730575,2196040,
%U A350286 2760736,3440800,4254250,5221125,6363630,7706286,9276085,11102650,13218400,15658720,18462136,21670495,25329150
%N A350286 Number of different ways to partition the set of vertices of a convex (n+11)-gon into 4 nonintersecting polygons.
%C A350286 Equivalently, the number of noncrossing set partitions of an (n+11)-set into 4 blocks with 3 or more elements in each block.
%H A350286 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A350286 a(n) = (n*(n+1)*(n+2)*(n+9)*(n+10)*(n+11))/144.
%F A350286 G.f.: x*(55 - 99*x + 63*x^2 - 14*x^3)/(1 - x)^7. - _Stefano Spezia_, Dec 26 2021
%e A350286 a(1) = 55; solutions are {1,2,3} {4,5,6} {7,8,9} {10,11,12} with 3 different orientations, {1,2,3} {4,5,6} {11,12,7} {8,9,10} with 12 different orientations, {1,2,3} {12,4,5} {11,6,7} {8,9,10} with 12 different orientations, {1,2,3} {12,4,5} {10,11,6} {7,8,9} with 12 different orientations, {1,2,3} {4,5,6} {12,7,8} {9,10,11} with 12 orientations and {1,2,3} {4,8,12} {5,6,7} {9,10,11} with 4 orientations.
%e A350286 The above numbers can be considered to be the partition of a 12-set into 4 blocks or the partition of the vertices of a convex 12-gon into 4 triangles with vertices labeled 1,2,3,...,12 in order.
%e A350286 a(2) = 286 corresponding to the number of different ways to partition the vertices of a 13-gon into three triangles and one quadrilateral.
%t A350286 a[n_] := n*(n + 1)*(n + 2)*(n + 9)*(n + 10)*(n + 11)/144; Array[a, 35, 0] (* _Amiram Eldar_, Dec 26 2021 *)
%Y A350286 Column k=4 of A350248.
%Y A350286 Cf. A350116.
%K A350286 easy,nonn
%O A350286 0,2
%A A350286 _Janaka Rodrigo_, Dec 23 2021