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%I #7 Jun 01 2015 06:28:50
%S 1430,175032,12597000,698377680,33079524324,1411221754800,
%T 55928745100800,2100173331484800,75727786603836510,
%U 2646827388046104120,90290940344491887000,3021580012515765901200,99583828881536195805180,3242049884573075122369680
%N Number of 2n-length strings of balanced parentheses of exactly 8 different types that are introduced in ascending order.
%H Alois P. Heinz, <a href="/A258396/b258396.txt">Table of n, a(n) for n = 8..650</a>
%F Recurrence: (n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*n*(n+1)*a(n) = 72*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*n*(2*n - 1)*a(n-1) - 2184*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(2*n - 3)*(2*n - 1)*a(n-2) + 36288*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-3) - 359184*(n-6)*(n-5)*(n-4)*(n-3)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-4) + 2153088*(n-6)*(n-5)*(n-4)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-5) - 7559936*(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) + 14026752*(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) - 10321920*(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). - _Vaclav Kotesovec_, Jun 01 2015
%F a(n) ~ 32^n / (8!*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Jun 01 2015
%Y Column k=8 of A253180.
%K nonn
%O 8,1
%A _Alois P. Heinz_, May 28 2015
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