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%I #25 Jan 31 2020 15:16:45
%S 0,1,2,6,24,108,516,2556,12972,66996,350628,1854252,9888924,53107236,
%T 286882740,1557510012,8492587596,46483203348,255273601860,
%U 1406078670924,7765563869436,42991470093060,238528474655316,1326059132006556
%N Number of permutations of length n avoiding each length 5 pattern p with p(1)=5 and (p(5)=4 or p(4)=4).
%H Vincenzo Librandi, <a href="/A178594/b178594.txt">Table of n, a(n) for n = 0..200</a>
%H Elena Barcucci, Vincent Vajnovszki, <a href="https://doi.org/10.1016/j.tcs.2012.02.039">Generalized Schroeder permutations</a>, Theoretical Computer Science, Volume 502, 2 September 2013, Pages 210-216.
%F Generating function: (1+2*x+3*x*(-x+1-(1-6*x+x^2)^(1/2))/(x+(1-6*x+x^2)^(1/2)))*x.
%F Recurrence (for n>=5): (n-2)*a(n) = 3*(4*n - 11)*a(n-1) - (37*n - 131)*a(n-2) + 6*(n-5)*a(n-3). - _Vaclav Kotesovec_, Nov 20 2012
%F a(n) ~ 3*sqrt(3*sqrt(2)-4)*(3+2*sqrt(2))^n/(n^(3/2)*sqrt(Pi)). - _Vaclav Kotesovec_, Nov 20 2012
%t CoefficientList[Series[x(1 + 2x + 3x(-x + 1 - (1 - 6x + x^2)^(1/2))/(x + (1 - 6x + x^2)^(1/2))), {x, 0, 23}], x] (* _Robert G. Wilson v_, Jun 28 2010 *)
%o (PARI) x='x+O('x^50); concat([0], Vec((1+2*x+3*x*(-x+1-(1-6*x+x^2)^(1/2))/(x+(1-6*x+x^2)^(1/2)))*x)) \\ _G. C. Greubel_, Mar 24 2017
%K nonn
%O 0,3
%A Vincent Vajnovszki (vvajnov(AT)u-bourgogne.fr), May 30 2010
%E a(19) onward from _Robert G. Wilson v_, Jun 28 2010
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