# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a121704 Showing 1-1 of 1 %I A121704 #23 Nov 05 2018 04:10:18 %S A121704 1,2,4,10,24,64,166,456,1234,3454,9600,27246,77132,221336,635078, %T A121704 1839000,5331274,15555586,45465412,133517130,392841336,1160033656, %U A121704 3432015726,10182891552,30267591290,90177226062,269117947728,804699330974,2409839825756,7228746487536 %N A121704 Number of separable involutions. %C A121704 The separable permutations are those avoiding 2413 and 3142 and are counted by the large Schroeder numbers (A006318). %C A121704 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 A121704 Alois P. Heinz, Table of n, a(n) for n = 1..600 %H A121704 Miklós Bóna, Cheyne Homberger, Jay Pantone, and Vince Vatter, Pattern-avoiding involutions: exact and asymptotic enumeration, arxiv:1310.7003 [math.CO], 2013-2014. %H A121704 R. Brignall, S. Huczynska and V. Vatter, Simple permutations and algebraic generating functions, arXiv:math/0608391 [math.CO], 2006. %F A121704 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 A121704 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 A121704 a(5) = 24 because of the 26 involutions of length 5 only two are not separable, 35142 and 42513. %t A121704 terms = 30; %t A121704 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 A121704 CoefficientList[f[x]/x, x] (* _Jean-François Alcover_, Nov 05 2018 *) %Y A121704 Cf. A121703. %K A121704 nonn %O A121704 1,2 %A A121704 _Vincent Vatter_, Aug 16 2006 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE