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A352900 a(n) is the number of different ways to partition the set of vertices of a convex n-gon into intersecting polygons.

%I A352900 #54 Aug 04 2025 07:47:06
%S A352900 0,0,0,7,28,79,460,2486,11209,59787,361777,2167635,13577211,91919186,
%T A352900 650059294,4761980740,36508824672,292116858616,2424047807182,
%U A352900 20847409357919,185754041370693,1711253802075941,16272637412753211,159561718074359537,1611599794862761838,16747401536440092104
%N A352900 a(n) is the number of different ways to partition the set of vertices of a convex n-gon into intersecting polygons.
%F A352900 a(n) = A006505(n) - A114997(n).
%F A352900 a(n) = Sum_{k=2..floor(n/3)} (T(n,k) - C(n+1,k)*C(n-2k-1,k-1)/(n+1)); n > 5, where T(n,k) = k*T(n-1,k) + C(n-1,2)*T(n-3,k-1); n > 5 and 1 < k <= floor(n/3), T(n,k) = 1 when k = 1.
%F A352900 T(n,k) = A059022(n,k) is the number of different ways to partition the set of vertices of a convex n-gon into k polygons.
%e A352900 For n=6, there are a(6) = 7 intersecting partitions of the convex hexagon. On vertices 1..6, they are the following pairs of triangles:
%e A352900   {1,3,4}, {5,6,2}
%e A352900   {4,5,1}, {2,3,6}
%e A352900   {3,4,6}, {1,2,5}
%e A352900   {2,3,5}, {1,4,6}
%e A352900   {1,2,4}, {5,6,3}
%e A352900   {1,6,3}, {5,4,2}
%e A352900   {1,3,5}, {2,4,6}
%o A352900 (PARI) T2(n,k) = if (n<3, 0, if (k==1, 1, k*T2(n-1,k) + binomial(n-1,2)*T2(n-3,k-1))); \\ A059022
%o A352900 a5(n) = if (n<3, n==0, sum(k=1, n\3, T2(n,k))); \\ A006505
%o A352900 a7(n) = sum(k=ceil((n+3)/2), n, (1/(n+1) * binomial(n+1, k) * binomial(2*k-n-3, n-k)) ); \\ A114997
%o A352900 a(n) =  a5(n) - a7(n); \\ _Michel Marcus_, Apr 09 2022
%Y A352900 Cf. A006505, A059022, A114997, A350248.
%K A352900 nonn
%O A352900 3,4
%A A352900 _Janaka Rodrigo_, Apr 07 2022
%E A352900 More terms from _Michel Marcus_, Apr 09 2022