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A352477 a(n) is the number of different ways to partition the set of vertices of a convex n-gon into 4 intersecting polygons.

%I A352477 #26 Apr 03 2022 08:54:02
%S A352477 15345,199914,1610700,10333050,57958005,297787980,1439757200,
%T A352477 6662364668,29844152321,130445708134,559533869356,2365296230374,
%U A352477 9885290683829,40944327268760,168389163026240,688631375953560,280357073972529,11373212442818370,46006062638648940
%N A352477 a(n) is the number of different ways to partition the set of vertices of a convex n-gon into 4 intersecting polygons.
%F A352477 a(n) = A261724(n) - A350286(n - 11), n > 11.
%e A352477 The set of vertices of a convex 14-gon can be partitioned into 4 polygons in 1611610 different ways:
%e A352477 - 3 triangles and 1 pentagon in (1/3!)*C(14,3)*C(11,3)*C(8,3)*C(5,5) = 560560 different ways, and
%e A352477 - 2 triangles and 2 quadrilaterals in (1/2!)*(1/2!)*C(14,3)*C(11,3)*C(8,4)*C(4,4) = 1051050 ways.
%e A352477 Subtracting the A350286(14-11)=910 nonintersecting partitions leaves a(14)=1610700.
%o A352477 (PARI) a4(n) = (1/12)*(-3^(n - 2)*(n^2 + 5*n + 18) + (1/64)*(2^(2*n + 5) + 3*2^n*(n^4 + 2*n^3 + 19*n^2 + 42*n + 64) - 16*(n^6 - 9*n^5 + 43*n^4 - 91*n^3 + 112*n^2 - 32*n + 8))); \\ A261724
%o A352477 a6(n) = (n*(n+1)*(n+2)*(n+9)*(n+10)*(n+11))/144; \\ A350286
%o A352477 a(n) = a4(n) - a6(n-11); \\ _Michel Marcus_, Mar 20 2022
%Y A352477 Cf. A261724, A350116, A350286.
%K A352477 nonn
%O A352477 12,1
%A A352477 _Janaka Rodrigo_, Mar 17 2022