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%I #8 Jun 01 2015 05:51:01
%S 2,15,98,630,4092,27027,181610,1239810,8582756,60138078,425800564,
%T 3042175500,21906338040,158830645635,1158564772890,8496271312650,
%U 62604582047700,463275674416170,3441483002640540,25654715940496500,191852749820189640,1438895966711035950
%N Number of 2n-length strings of balanced parentheses of exactly 2 different types that are introduced in ascending order.
%H Alois P. Heinz, <a href="/A258390/b258390.txt">Table of n, a(n) for n = 2..1000</a>
%F a(n) = (2*n-1)*(6*n*a(n-1)-8*(2*n-3)*a(n-2))/(n*(n+1)) for n>2, a(2)=2, a(n)=0 for n<2.
%F a(n) = (2^(n-1)-1) * binomial(2n,n)/(n+1) = (2^(n-1)-1)*A000108(n). - _Vaclav Kotesovec_, Jun 01 2015
%p a:= proc(n) option remember; `if`(n<3, [0$2, 2][n+1],
%p (2*n-1)*(6*n*a(n-1) -8*(2*n-3)*a(n-2))/(n*(n+1)))
%p end:
%p seq(a(n), n=2..25);
%t Table[(2^(n-1)-1)*Binomial[2n,n]/(n+1),{n,2,20}] (* _Vaclav Kotesovec_, Jun 01 2015 *)
%Y Column k=2 of A253180.
%K nonn
%O 2,1
%A _Alois P. Heinz_, May 28 2015
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