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%I #7 Jun 01 2015 06:09:11
%S 42,1980,60060,1501500,33795762,714249900,14504269780,286931752800,
%T 5578065392900,107178276605400,2043352620527400,38758743724018500,
%U 732849800716048290,13831507110589591500,260829110106412824900,4917878997439418010000,92758042341429880435020
%N Number of 2n-length strings of balanced parentheses of exactly 5 different types that are introduced in ascending order.
%H Alois P. Heinz, <a href="/A258393/b258393.txt">Table of n, a(n) for n = 5..750</a>
%F Recurrence: (n-3)*(n-2)*(n-1)*n*(n+1)*a(n) = 30*(n-3)*(n-2)*(n-1)*n*(2*n - 1)*a(n-1) - 340*(n-3)*(n-2)*(n-1)*(2*n - 3)*(2*n - 1)*a(n-2) + 1800*(n-3)*(n-2)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-3) - 4384*(n-3)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-4) + 3840*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-5). - _Vaclav Kotesovec_, Jun 01 2015
%F a(n) ~ 20^n / (120*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Jun 01 2015
%Y Column k=5 of A253180.
%K nonn
%O 5,1
%A _Alois P. Heinz_, May 28 2015
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