OFFSET
3,4
FORMULA
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.
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.
EXAMPLE
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:
{1,3,4}{5,6,2}
{4,5,1}{2,3,6}
{3,4,6}{1,2,5}
{2,3,5}{1,4,6}
{1,2,4}{5,6,3}
{1,6,3}{5,4,2}
{1,3,5}{2,4,6}
PROG
(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
a5(n) = if (n<3, n==0, sum(k=1, n\3, T2(n, k))); \\ A006505
a7(n) = sum(k=ceil((n+3)/2), n, (1/(n+1) * binomial(n+1, k) * binomial(2*k-n-3, n-k)) ); \\ A114997
a(n) = a5(n) - a7(n); \\ Michel Marcus, Apr 09 2022
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Janaka Rodrigo, Apr 07 2022
EXTENSIONS
More terms from Michel Marcus, Apr 09 2022
STATUS
approved