A352900 - OEIS

%I #50 Apr 21 2022 14:01:21

%S 0,0,0,7,28,79,460,2486,11209,59787,361777,2167635,13577211,91919186,

%T 650059294,4761980740,36508824672,292116858616,2424047807182,

%U 20847409357919,185754041370693,1711253802075941,16272637412753211,159561718074359537,1611599794862761838,16747401536440092104

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

%F a(n) = A006505(n) - A114997(n).

%F 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 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 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   {1,3,4}{5,6,2}

%e   {4,5,1}{2,3,6}

%e   {3,4,6}{1,2,5}

%e   {2,3,5}{1,4,6}

%e   {1,2,4}{5,6,3}

%e   {1,6,3}{5,4,2}

%e   {1,3,5}{2,4,6}

%o (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 a5(n) = if (n<3, n==0, sum(k=1, n\3, T2(n,k))); \\ A006505

%o 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 a(n) =  a5(n) - a7(n); \\ _Michel Marcus_, Apr 09 2022

%Y Cf. A006505, A114997, A059022, A350248.

%K nonn,more

%O 3,4

%A _Janaka Rodrigo_, Apr 07 2022

%E More terms from _Michel Marcus_, Apr 09 2022