A145868 Number of involutions of length 2n+1 which are invariant under the reverse-complement map and have no decreasing subsequences of length 7.
1, 2, 6, 19, 68, 255, 1020, 4221, 18186, 80304, 364476, 1684782, 7944156, 37988379, 184406508, 905147815, 4495346570, 22527055980, 113957354940, 580759868910, 2982724210440, 15414453711930, 80177422383240, 419249099692710, 2204316120027420, 11642676960438000
Offset: 0
Keywords
Programs
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Mathematica
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}]
Formula
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).
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
a(n) ~ 6^(n+7/2) / (2 * Pi^(3/2) * n^(7/2)). - Vaclav Kotesovec, Feb 18 2015
Extensions
More terms from Vaclav Kotesovec, Feb 18 2015