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%I #7 Jun 01 2015 05:58:48
%S 5,84,1050,11880,129129,1381380,14707550,156706680,1675459786,
%T 17998446312,194361212500,2110052926800,23026236054345,
%U 252513376831620,2781895215981750,30778564965687000,341873708072702190,3811170628172227320,42628644369844747500
%N Number of 2n-length strings of balanced parentheses of exactly 3 different types that are introduced in ascending order.
%H Alois P. Heinz, <a href="/A258391/b258391.txt">Table of n, a(n) for n = 3..900</a>
%F Recurrence: (n-1)*n*(n+1)*a(n) = 12*(n-1)*n*(2*n - 1)*a(n-1) - 44*(n-1)*(2*n - 3)*(2*n - 1)*a(n-2) + 48*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-3). - _Vaclav Kotesovec_, Jun 01 2015
%F a(n) ~ 12^n / (6*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Jun 01 2015
%Y Column k=3 of A253180.
%K nonn
%O 3,1
%A _Alois P. Heinz_, May 28 2015
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