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%I #33 Feb 07 2022 00:25:51
%S 1,2,5,13,30,61,112,190,303,460,671,947,1300,1743,2290,2956,3757,4710,
%T 5833,7145,8666,10417,12420,14698,17275,20176,23427,27055,31088,35555,
%U 40486,45912,51865,58378,65485,73221,81622,90725,100568,111190,122631,134932
%N Number of permutations of length n which avoid the patterns 123 and 4312.
%C Also number of permutations of length n which avoid the patterns 321, 2134 (reverse symmetry); or 321, 1243 (complement symmetry); etc.
%H Vincenzo Librandi, <a href="/A116699/b116699.txt">Table of n, a(n) for n = 1..1000</a>
%H Christian Bean, Bjarki Gudmundsson, Henning Ulfarsson, <a href="https://arxiv.org/abs/1705.04109">Automatic discovery of structural rules of permutation classes</a>, arXiv:1705.04109 [math.CO], 2017.
%H Lara Pudwell, <a href="http://faculty.valpo.edu/lpudwell/maple/webbook/bookmain.html">Systematic Studies in Pattern Avoidance</a>, 2005.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F G.f.: x*(2*x^3 - 5*x^2 + 3*x - 1)/(x-1)^5.
%F a(n) = (n^4 + 2*n^3 - 13*n^2 + 34*n)/24. - _Franklin T. Adams-Watters_, Sep 16 2006
%F Partial sums of A105163. - Levi R. Self (levi.r.self(AT)gmail.com), Aug 04 2007
%F Binomial transform of [1, 1, 2, 3, 1, 0, 0, 0, ...]. - _Gary W. Adamson_, Oct 23 2007
%t LinearRecurrence[{5,-10,10,-5,1}, {1,2,5,13,30}, 50] (* _Vladimir Joseph Stephan Orlovsky_, Feb 02 2012 *)
%t CoefficientList[Series[(2 x^3 - 5 x^2 + 3 x - 1)/(x - 1)^5, {x, 0, 40}], x] (* _Vincenzo Librandi_, Nov 01 2014 *)
%o (PARI) for(n=1,100,print1((n^4 + 2*n^3 - 13*n^2 + 34*n)/24",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 22 2008
%o (Magma) [(n^4 + 2*n^3 - 13*n^2 + 34*n)/24: n in [1..45]]; // _Vincenzo Librandi_, Nov 01 2014
%K nonn,easy
%O 1,2
%A _Lara Pudwell_, Feb 26 2006
%E Edited by _N. J. A. Sloane_, Mar 16 2008
%E More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 22 2008
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