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%I #14 Jun 14 2016 10:27:46
%S 1,1,3,12,51,217,912,3785,15554,63458,257566,1041548,4200462,16906262,
%T 67943341,272740788,1093881967,4384217569,17562176283,70319782015,
%U 281466691159,1126304935761,4505961410365,18023526090613,72082118816829,288245594631227,1152536796877409,4607992736095739,18422141293792669,73645313049839723
%N The number of sum indecomposable permutations which avoid the patterns 3124 and 4312.
%H Vincenzo Librandi, <a href="/A228770/b228770.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.: -(24*x^6 - 71*x^5 + 84*x^4 - 45*x^3 + 11*x^2 + sqrt(-4*x + 1)*(4*x^6 - 25*x^5 + 40*x^4 - 29*x^3 + 9*x^2 - x) - x)/(8*x^6 - 54*x^5 + 117*x^4 - 114*x^3 + 54*x^2 - sqrt(-4*x + 1)*(12*x^5 - 43*x^4 + 58*x^3 - 36*x^2 + 10*x - 1) - 12*x + 1).
%F a(n) ~ 2^(2*n-1)/9 * (1+2/(sqrt(Pi*n))). - _Vaclav Kotesovec_, Mar 20 2014
%F Conjecture: -(n+1)*(39961*n-2474598)*a(n) +(-39961*n^2-25975201*n+4949196) *a(n-1) +3*(1460811*n^2+27429105*n-41310802) *a(n-2) +3 *(-8653921*n^2-4750029*n+74360724) *a(n-3) +4*(15005713*n^2-82481258*n+83094771) *a(n-4) +12*(-4937548*n^2+40726604*n-73155719) *a(n-5) +16*(652718*n-2110173)*(2*n-13) *a(n-6)=0. - _R. J. Mathar_, Jun 14 2016
%e Example: a(4)=12 because there are 12 sum indecomposable permutations of length 4 which avoid the patterns 3124 and 4312.
%t CoefficientList[Series[- (1/x) (24 x^6 - 71 x^5 + 84 x^4 - 45 x^3 + 11 x^2 + Sqrt[-4 x + 1] (4 x^6 - 25 x^5 + 40 x^4 - 29 x^3 + 9 x^2 - x) - x) / (8 x^6 - 54 x^5 + 117 x^4 - 114 x^3 + 54 x^2 - Sqrt[-4 x + 1] (12 x^5 - 43 x^4 + 58 x^3 - 36 x^2 + 10 x - 1) - 12 x + 1), {x, 0, 30}], x] (* _Vincenzo Librandi_, Sep 09 2013 *)
%Y A228770(n) = A165534(n) - A226434(n)
%K nonn
%O 1,3
%A _Jay Pantone_, Sep 08 2013
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