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%I #7 Jun 01 2015 06:25:49
%S 429,40040,2246244,98760480,3761539782,130505896752,4245988489600,
%T 131928199603200,3962683868528385,116039722090972680,
%U 3332921846278964940,94315723869947580000,2638390752595156276410,73147630662437905413840,2013841857892713303414960
%N Number of 2n-length strings of balanced parentheses of exactly 7 different types that are introduced in ascending order.
%H Alois P. Heinz, <a href="/A258395/b258395.txt">Table of n, a(n) for n = 7..650</a>
%F Recurrence: (n-5)*(n-4)*(n-3)*(n-2)*(n-1)*n*(n+1)*a(n) = 56*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*n*(2*n - 1)*a(n-1) - 1288*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(2*n - 3)*(2*n - 1)*a(n-2) + 15680*(n-5)*(n-4)*(n-3)*(n-2)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-3) - 108304*(n-5)*(n-4)*(n-3)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-4) + 420224*(n-5)*(n-4)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-5) - 836352*(n-5)*(2*n - 11)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-6) + 645120*(2*n - 13)*(2*n - 11)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-7). - _Vaclav Kotesovec_, Jun 01 2015
%F a(n) ~ 28^n / (7!*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Jun 01 2015
%Y Column k=7 of A253180.
%K nonn
%O 7,1
%A _Alois P. Heinz_, May 28 2015
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