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%I #23 Nov 05 2018 04:10:18
%S 1,2,4,10,24,64,166,456,1234,3454,9600,27246,77132,221336,635078,
%T 1839000,5331274,15555586,45465412,133517130,392841336,1160033656,
%U 3432015726,10182891552,30267591290,90177226062,269117947728,804699330974,2409839825756,7228746487536
%N Number of separable involutions.
%C The separable permutations are those avoiding 2413 and 3142 and are counted by the large Schroeder numbers (A006318).
%C The involutions avoiding 2413 coincide with the involutions avoiding 3142, and hence both sets coincide with the separable involutions. - _David Callan_, Aug 27 2014
%H Alois P. Heinz, <a href="/A121704/b121704.txt">Table of n, a(n) for n = 1..600</a>
%H Miklós Bóna, Cheyne Homberger, Jay Pantone, and Vince Vatter, <a href="http://arxiv.org/abs/1310.7003">Pattern-avoiding involutions: exact and asymptotic enumeration</a>, arxiv:1310.7003 [math.CO], 2013-2014.
%H R. Brignall, S. Huczynska and V. Vatter, <a href="https://arxiv.org/abs/math/0608391">Simple permutations and algebraic generating functions</a>, arXiv:math/0608391 [math.CO], 2006.
%F G.f. f satisfies: x^2f^4 + (x^3+3x^2+x-1)f^3 + (3x^3+6x^2-x)f^2 + (3x^3+7x^2-x-1)f +x^3+3x^2+x=0.
%F a(n) ~ sqrt(6 + 6*sqrt(2) + 4*sqrt(3) + 3*sqrt(6)) * (5+2*sqrt(6))^(n/2) / (2 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Sep 13 2014
%e a(5) = 24 because of the 26 involutions of length 5 only two are not separable, 35142 and 42513.
%t terms = 30;
%t f[_] = 0; Do[f[x_] = Normal[(-(x^3 f[x]^3) - 3 x^3 f[x]^2 - x^2 f[x]^4 - 3 x^2 f[x]^3 - 6 x^2 f[x]^2 - x f[x]^3 + f[x]^3 + x f[x]^2 - x^3 - 3 x^2 - x)/(3 x^3 + 7 x^2 - x - 1) + O[x]^(terms+1)], {terms+1}];
%t CoefficientList[f[x]/x, x] (* _Jean-François Alcover_, Nov 05 2018 *)
%Y Cf. A121703.
%K nonn
%O 1,2
%A _Vincent Vatter_, Aug 16 2006
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