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Number of permutations of length 2n which are invariant under the reverse-complement map and have no decreasing subsequences of length 5.
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%I #12 Feb 18 2015 18:10:30

%S 1,2,8,46,332,2784,25888,259382,2749244,30449416,349379648,4127103776,

%T 49954287424,617299996928,7765434294912,99214734136966,

%U 1285011754097372,16845342401817048,223216584359771296,2986529546579794040,40308007404730514096,548337251596355725312

%N Number of permutations of length 2n which are invariant under the reverse-complement map and have no decreasing subsequences of length 5.

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

%F Recurrence: (n+1)^2*(n+2)*(3*n-1)*a(n) = 2*(30*n^4 + 11*n^3 - 20*n^2 - 3*n + 6)*a(n-1) - 64*(n-1)^3*(3*n+2)*a(n-2). - _Vaclav Kotesovec_, Feb 18 2015

%F a(n) ~ 2^(4*n+3) / (Pi^(3/2) * n^(7/2)). - _Vaclav Kotesovec_, Feb 18 2015

%e a(4) = 1*1*14 + 16*1*5 + 36*2*2 + 16*5*1 + 1*14*1 = 332.

%t Table[Sum[ Binomial[n, j]^2*Binomial[2*j, j]* Binomial[2*(n - j), n - j]/((n - j + 1)*(j + 1)), {j, 0, n}], {n, 0, 20}]

%K nonn

%O 0,2

%A _Eric S. Egge_, Oct 21 2008

%E More terms from _Vaclav Kotesovec_, Feb 18 2015