%I #11 Feb 18 2015 18:21:14

%S 1,2,6,19,68,255,1020,4221,18186,80304,364476,1684782,7944156,

%T 37988379,184406508,905147815,4495346570,22527055980,113957354940,

%U 580759868910,2982724210440,15414453711930,80177422383240,419249099692710,2204316120027420,11642676960438000

%N Number of involutions of length 2n+1 which are invariant under the reverse-complement map and have no decreasing subsequences of length 7.

%F a(n) = sum(j,0,n, C(n,j)*C(n-j,floor((n-j)/2))*A000108(floor((j+1)/2)) * A000108(ceiling((j+1)/2))), where C(n,j) = n!/(j!(n-j)!) and A000108(n) = Catalan(n).

%F Recurrence: (n+3)*(n+4)*(n+5)*(2*n+1)*(2*n+3)*a(n) = 8*(2*n+1)*(5*n^3 + 33*n^2 + 67*n + 45)*a(n-1) + 4*(n-1)*(40*n^4 + 216*n^3 + 326*n^2 + 144*n + 45)*a(n-2) - 288*(n-2)*(n-1)*(n+1)*(2*n+5)*a(n-3) - 144*(n-3)*(n-2)*(n-1)*(2*n+3)*(2*n+5)*a(n-4). - _Vaclav Kotesovec_, Feb 18 2015

%F a(n) ~ 6^(n+7/2) / (2 * Pi^(3/2) * n^(7/2)). - _Vaclav Kotesovec_, Feb 18 2015

%t Array[Cat, 21, 0]; For[i = 0, i < 21, ++i, Cat[i] = (1/(i + 1))*Binomial[2*i, i]]; Table[Sum[ Binomial[n, j]*Binomial[n - j, Floor[(n - j)/2]]* Cat[Floor[(j + 1)/2]]*Cat[Ceiling[(j + 1)/2]], {j, 0, n}], {n, 0, 15}]

%K nonn

%O 0,2

%A _Eric S. Egge_, Oct 22 2008

%E More terms from _Vaclav Kotesovec_, Feb 18 2015