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%I #13 Mar 18 2014 11:10:43
%S 0,1,3,10,35,129,494,1935,7670,30582,122280,489552,1960956,7855994,
%T 31471731,126063782,504888839,2021777865,8094784697,32405289263,
%U 129709206465,519129580361,2077477804103,8313000733125,33261722967167,133076495664483,532391828669675,2129796460981743,8519701993370619,34079469569317323
%N The number of skew sum decomposable permutations which avoid the patterns 3124 and 4312.
%H Vincenzo Librandi, <a href="/A228769/b228769.txt">Table of n, a(n) for n = 1..1000</a>
%H Jay Pantone, <a href="http://arxiv.org/abs/1309.0832">The Enumeration of Permutations Avoiding 3124 and 4312</a>, arXiv:1309.0832 [math.CO], (2013)
%F G.f.: -(3*x^4 - x^3 + sqrt(-4*x + 1)*(4*x^5 - 9*x^4 + 9*x^3 - 2*x^2))/(12*x^4 - 31*x^3 + 27*x^2 + sqrt(-4*x + 1)*(4*x^4 - 13*x^3 + 15*x^2 - 7*x + 1) - 9*x + 1).
%F a(n) ~ 4^(n-1)/9 * (1 + 1/sqrt(Pi*n)). - _Vaclav Kotesovec_, Mar 18 2014
%e Example: a(4)=10 because there are 10 skew sum decomposable permutations of length 4 which avoid the patterns 3124 and 4312.
%t CoefficientList[Series[- (1/x) (3 x^4 - x^3 + Sqrt[-4 x + 1] (4 x^5 - 9 x^4 + 9 x^3 - 2 x^2)) / (12 x^4 - 31 x^3 + 27 x^2 + Sqrt[-4 x + 1] (4 x^4 - 13 x^3 + 15 x^2 - 7 x + 1) - 9 x + 1), {x, 0, 40}], x] (* _Vincenzo Librandi_, Sep 09 2013 *)
%Y The class of all permutations which avoid the patterns 3124 and 4312 is given by A165534.
%K nonn
%O 1,3
%A _Jay Pantone_, Sep 08 2013
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