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A228771 The number of skew sum indecomposable permutations which avoid the patterns 3124 and 4312.

%I A228771 #12 Jun 14 2016 10:30:01
%S A228771 1,1,3,12,53,234,1013,4306,18051,74903,308487,1263393,5152139,
%T A228771 20941298,84897207,343467388,1387244237,5595368133,22543241377,
%U A228771 90739796783,364954106877,1466865660103,5892463315373,23658818086719,94952826295865,380947979933041,1527871081396065,6126157580638517,24557525359295337,98421154766829972
%N A228771 The number of skew sum indecomposable permutations which avoid the patterns 3124 and 4312.
%H A228771 Vincenzo Librandi, <a href="/A228771/b228771.txt">Table of n, a(n) for n = 1..1000</a>
%H A228771 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 A228771 G.f.: (8*x^6 - 28*x^5 + 50*x^4 - 35*x^3 + 10*x^2 - sqrt(-4*x + 1)*(6*x^5 - 18*x^4 + 21*x^3 - 8*x^2 + x) - x)/(8*x^5 - 46*x^4 + 71*x^3 - 43*x^2 - sqrt(-4*x + 1)*(12*x^4 - 31*x^3 + 27*x^2 - 9*x + 1) + 11*x - 1).
%F A228771 a(n) ~ 4^(n-1)/3 * (1+1/sqrt(Pi*n)). - _Vaclav Kotesovec_, Mar 20 2014
%F A228771 Conjecture: -163*(n+2)*(4*n-413) *a(n) +(-652*n^2-725425*n-452889) *a(n-1) +5*(14473*n^2+512276*n-443094) *a(n-2) +(-410045*n^2-2408964*n+8429009) *a(n-3) +2*(404156*n^2-1297075*n-1518084)*a(n-4) -8*(29333*n-32490)*(2*n-11)*a(n-5)=0. - _R. J. Mathar_, Jun 14 2016
%e A228771 Example: a(4)=12 because there are 12 skew sum indecomposable permutations of length 4 which avoid the patterns 3124 and 4312.
%t A228771 CoefficientList[Series[(1/x) (8 x^6 - 28 x^5 + 50 x^4 - 35 x^3 + 10 x^2 - Sqrt[-4 x + 1] (6 x^5 - 18 x^4 + 21 x^3 - 8 x^2 + x) - x) / (8 x^5 - 46 x^4 + 71 x^3 - 43 x^2 - Sqrt[-4 x + 1] (12 x^4 - 31 x^3 + 27 x^2 - 9 x + 1) + 11 x - 1), {x, 0, 30}], x] (* _Vincenzo Librandi_, Sep 09 2013 *)
%Y A228771 A228771(n) = A165534(n) - A228769(n)
%K A228771 nonn
%O A228771 1,3
%A A228771 _Jay Pantone_, Sep 08 2013
%E A228771 Corrected a(17) by _Vincenzo Librandi_, Sep 09 2013