This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A165538 #53 Jul 06 2024 13:21:24
%S A165538 1,1,2,6,22,88,367,1568,6810,29943,132958,595227,2683373,12170778,
%T A165538 55499358,254297805,1170248190,5406570910,25068420955,116617923611,
%U A165538 544157590706,2546278167018,11945937322413,56180864428301,264812677643417,1250853429148333,5920145717412047
%N A165538 Number of permutations of length n which avoid the patterns 4312 and 3142.
%H A165538 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>, arXiv:1209.0425 [math.CO], 2012.
%H A165538 Christian Bean, <a href="https://hdl.handle.net/20.500.11815/1184">Finding structure in permutation sets</a>, Ph.D. Dissertation, Reykjavík University, School of Computer Science, 2018.
%H A165538 Christian Bean, Émile Nadeau, Henning Ulfarsson, <a href="https://arxiv.org/abs/1912.07503">Enumeration of Permutation Classes and Weighted Labelled Independent Sets</a>, arXiv:1912.07503 [math.CO], 2019.
%H A165538 Robert Brignall, Jakub Sliacan, <a href="https://arxiv.org/abs/1611.05370">Juxtaposing Catalan permutation classes with monotone ones</a>, arXiv:1611.05370 [math.CO], 2016.
%H A165538 Juan B. Gil, Michael D. Weiner, <a href="https://arxiv.org/abs/1812.01682">On pattern-avoiding Fishburn permutations</a>, arXiv:1812.01682 [math.CO], 2018.
%H A165538 Darla Kremer and Wai Chee Shiu, <a href="http://dx.doi.org/10.1016/S0012-365X(03)00042-6">Finite transition matrices for permutations avoiding pairs of length four patterns</a>, Discrete Math. 268 (2003), 171-183. MR1983276 (2004b:05006). See Table 1.
%H A165538 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 A165538 G.f. f satisfies: (x^3-2*x^2+x)*f^4+(4*x^3-9*x^2+6*x-1)*f^3+(6*x^3-12*x^2+7*x-1)*f^2+(4*x^3-5*x^2+x)*f+x^3 = 0.
%F A165538 From _Vaclav Kotesovec_, Jul 06 2024: (Start)
%F A165538 G.f.: (1 + sqrt(1-4*x)) / (4*x) - sqrt(2*(1 + sqrt(1-4*x)-2*x)*(1-x)*(1-5*x)) / (4*(1-x)*x).
%F A165538 a(n) ~ (1 + sqrt(5)) * 5^(n+1) / (16 * sqrt(Pi) * n^(3/2)). (End)
%e A165538 There are 22 permutations of length 4 which avoid these two patterns, so a(4)=22.
%t A165538 CoefficientList[Series[(1 + Sqrt[1 - 4*x]) / (4*x) - Sqrt[2*(1 + Sqrt[1 - 4*x] - 2*x)*(1 - x)*(1 - 5*x)] / (4*(1-x)*x), {x, 0, 30}], x] (* _Vaclav Kotesovec_, Jul 06 2024 *)
%K A165538 nonn
%O A165538 0,3
%A A165538 _Vincent Vatter_, Sep 21 2009
%E A165538 Reference corrected by _Vincent Vatter_, Sep 04 2012
%E A165538 a(0)=1 prepended by _Alois P. Heinz_, Jul 06 2024