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%I #7 Jun 01 2015 06:13:52
%S 132,9009,380380,12864852,383402292,10551322782,275335499824,
%T 6924802684800,169656773406120,4078556074277685,96700630711999860,
%U 2269529269318731420,52868514692841609300,1224857602490265215010,28265620407321158141280,650452332645092821924080
%N Number of 2n-length strings of balanced parentheses of exactly 6 different types that are introduced in ascending order.
%H Alois P. Heinz, <a href="/A258394/b258394.txt">Table of n, a(n) for n = 6..700</a>
%F Recurrence: (n-4)*(n-3)*(n-2)*(n-1)*n*(n+1)*a(n) = 42*(n-4)*(n-3)*(n-2)*(n-1)*n*(2*n - 1)*a(n-1) - 700*(n-4)*(n-3)*(n-2)*(n-1)*(2*n - 3)*(2*n - 1)*a(n-2) + 5880*(n-4)*(n-3)*(n-2)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-3) - 25984*(n-4)*(n-3)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-4) + 56448*(n-4)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-5) - 46080*(2*n - 11)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-6). - _Vaclav Kotesovec_, Jun 01 2015
%F a(n) ~ 24^n / (720*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Jun 01 2015
%Y Column k=6 of A253180.
%K nonn
%O 6,1
%A _Alois P. Heinz_, May 28 2015
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