%I #6 Jun 01 2015 06:32:48
%S 4862,755820,67897830,4633467300,267074035800,13733597077200,
%T 650800305634050,29021018652697500,1235362166419751370,
%U 50713478000403718500,2022835296688063807950,78843505678630977784500,3016017325414346802772080,113617986954086473298668800
%N Number of 2n-length strings of balanced parentheses of exactly 9 different types that are introduced in ascending order.
%H Alois P. Heinz, <a href="/A258397/b258397.txt">Table of n, a(n) for n = 9..650</a>
%F Recurrence: (n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*n*(n+1)*a(n) = 90*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*n*(2*n - 1)*a(n-1) - 3480*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(2*n - 3)*(2*n - 1)*a(n-2) + 75600*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-3) - 1012368*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-4) + 8618400*(n-7)*(n-6)*(n-5)*(n-4)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-5) - 46315520*(n-7)*(n-6)*(n-5)*(2*n - 11)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-6) + 150105600*(n-7)*(n-6)*(2*n - 13)*(2*n - 11)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-7) - 262803456*(n-7)*(2*n - 15)*(2*n - 13)*(2*n - 11)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-8) + 185794560*(2*n - 17)*(2*n - 15)*(2*n - 13)*(2*n - 11)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-9). - _Vaclav Kotesovec_, Jun 01 2015
%F a(n) ~ 36^n / (9!*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Jun 01 2015
%Y Column k=9 of A253180.
%K nonn
%O 9,1
%A _Alois P. Heinz_, May 28 2015