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%I #26 May 31 2017 23:01:15
%S 0,0,1,11,81,500,2794,14649,73489,356960,1691790,7864950,36000186,
%T 162697176,727505972,3223913365,14176874193,61926666824,268931341414,
%U 1161913686618,4997204887550,21404922261112,91351116184716,388581750349946,1647982988377786
%N Total number of 231 patterns in the set of permutations avoiding 123.
%C a(n) is the total number of 231 (and also 312) patterns in the set of all 123 avoiding n-permutations. Also the number of 231 (or 213, or 312) patterns in the set of all 132 avoiding n-permutations.
%H G. C. Greubel, <a href="/A210064/b210064.txt">Table of n, a(n) for n = 1..1000</a>
%H Cheyne Homberger, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v19i3p43">Expected patterns in permutation classes</a>, Electronic Journal of Combinatorics, 19(3) (2012), P43.
%F G.f.: x/(2*(1-4*x)^2) + (x-1)/(2*(1-4*x)^(3/2)) + 1/(2 - 8*x).
%F a(n) ~ n * 2^(2*n-3) * (1 - 6/sqrt(Pi*n)). - _Vaclav Kotesovec_, Mar 15 2014
%F Conjecture: n*(n-3)*a(n) +2*(-4*n^2+11*n-2)*a(n-1) +8*(n-1)*(2*n-3)*a(n-2)=0. - _R. J. Mathar_, Oct 08 2016
%e a(3) = 1 since there is only one 231 pattern in the set {132,213,231,312,321}.
%t Rest[CoefficientList[Series[x/(2*(1-4*x)^2) + (x-1)/(2*(1-4*x)^(3/2)) + 1/(2 - 8*x), {x, 0, 20}], x]] (* _Vaclav Kotesovec_, Mar 15 2014 *)
%o (PARI) x='x+O('x^50); concat([0,0], Vec(x/(2*(1-4*x)^2) + (x-1)/(2*(1-4*x)^(3/2)) + 1/(2 - 8*x))) \\ _G. C. Greubel_, May 31 2017
%Y Cf. A045720.
%K nonn
%O 1,4
%A _Cheyne Homberger_, Mar 16 2012
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