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%I #19 Sep 08 2022 08:45:48
%S 1,1,2,6,22,88,363,1508,6255,25842,106327,435965,1782733,7275351,
%T 29648647,120707058,491113791,1997372920,8121565606,33020039047,
%U 134248625367,545835561195,2219474787024,9025797884775,36709145207578,149320519008554,607466672855393
%N Number of permutations of length n which avoid the patterns 4231 and 3124.
%D Kremer, Darla and Shiu, Wai Chee; Finite transition matrices for permutations avoiding pairs of length four patterns. Discrete Math. 268 (2003), 171-183. MR1983276 (2004b:05006). See Table 1.
%H Alois P. Heinz, <a href="/A165535/b165535.txt">Table of n, a(n) for n = 0..1000</a>
%H M. H. Albert, M. D. Atkinson, and V. Vatter, <a href="http://arxiv.org/abs/1209.0425">Inflations of geometric grid classes: three case studies</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Enumerations_of_specific_permutation_classes#Classes_avoiding_two_patterns_of_length_4">Permutation classes avoiding two patterns of length 4</a>.
%F G.f.: 1+(1-8*x+20*x^2-20*x^3+10*x^4-2*x^5-(1-4*x+2*x^2)*sqrt(1-4*x)) / (2*(1-3*x+x^2)*(-1+5*x-4*x^2+x^3)).
%e There are 22 permutations of length 4 which avoid these two patterns, so a(4)=22.
%t CoefficientList[Series[1+(1-8*x+20*x^2-20*x^3+10*x^4-2*x^5-(1-4*x+ 2*x^2 )*Sqrt[1-4*x])/(2*(1-3*x+x^2)*(-1+5*x-4*x^2+x^3)), {x, 0, 30}], x] (* _G. C. Greubel_, Oct 22 2018 *)
%o (PARI) x='x+O('x^30); Vec(1+(1-8*x+20*x^2-20*x^3+10*x^4-2*x^5-(1-4*x +2*x^2)*sqrt(1-4*x))/(2*(1-3*x+x^2)*(-1+5*x-4*x^2+x^3))) \\ _G. C. Greubel_, Oct 22 2018
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!(1+(1-8*x+20*x^2-20*x^3+10*x^4-2*x^5-(1-4*x+2*x^2)*Sqrt(1-4*x)) / (2*(1-3*x+x^2)*(-1+5*x-4*x^2+x^3)))); // _G. C. Greubel_, Oct 22 2018
%K nonn,easy
%O 0,3
%A _Vincent Vatter_, Sep 21 2009
%E More terms, g.f., and reference by _Vincent Vatter_, Sep 04 2012
%E a(0)=1 prepended by _Alois P. Heinz_, Feb 18 2016
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