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A380591 a(n) is the number of dissections of a convex (n+2)-sided polygon by nonintersecting diagonals into triangles and quadrilaterals such that at least one of the dividing diagonals passes through a chosen vertex.

%I A380591 #32 Feb 08 2025 20:52:04
%S A380591 0,1,5,21,90,395,1773,8110,37686,177450,844935,4061762,19687020,
%T A380591 96107358,472132330,2332304055,11578595554,57736664825,289055592810,
%U A380591 1452381167325,7321620080550,37020073600755,187699184460450,954084756674088,4861008765722340
%N A380591 a(n) is the number of dissections of a convex (n+2)-sided polygon by nonintersecting diagonals into triangles and quadrilaterals such that at least one of the dividing diagonals passes through a chosen vertex.
%H A380591 Muhammed Sefa Saydam, <a href="/A380591/b380591.txt">Table of n, a(n) for n = 1..100</a>
%F A380591 a(n) = A001002(n) + A217596(n).
%F A380591 a(n) = -Sum_{i=0..n-2} A217596(i+1) * A001002(n-1-i), n >= 2.
%e A380591 a(2) = 1. Because: Let's choose vertex A in convex quadrilateral ABCD. At least one diagonal must pass through corner A. A diagonal is drawn at corner C. 2 triangles are obtained. There is 1 situation.
%e A380591 a(3) = 5. Because: Let's choose vertex A in convex pentagon ABCDE. At least one diagonal must pass through corner A. First case: diagonals AD and AC can be drawn. Second Case: Diagonals AD and DB can be drawn. Third case: Only diagonal AD can be drawn. Fourth Case: Diagonals AC and EC can be drawn. Fifth Case: Only diagonal AC can be drawn. There are 5 situations in total.
%p A380591 a:= proc(n) option remember; `if`(n<3, n*(n-1)/2, (n*(3059*n^2
%p A380591       -7876*n+4997)*a(n-1)+(4120*n^3-22681*n^2+39305*n-21644)*a(n-2)
%p A380591       +39*(n-3)*(3*n-10)*(3*n-11)*a(n-3))/(5*(n+1)*n*(142*n-337)))
%p A380591     end:
%p A380591 seq(a(n), n=1..25);  # _Alois P. Heinz_, Jan 27 2025
%o A380591 (PARI) a(n) = if(n==0,1,if(n==1,-1,-sum(i=ceil(n/2),n,binomial(i,n-i)*binomial(n+i-2,n-2)/(n-1)))) + (1/(n+1))*sum(k=ceil(n/2),n,binomial(n+k,k)*binomial(k,n-k)); \\ _Michel Marcus_, Jan 27 2025
%Y A380591 Cf. A001002, A217596.
%K A380591 nonn,easy
%O A380591 1,3
%A A380591 _Muhammed Sefa Saydam_, Jan 27 2025