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A117156 Number of permutations avoiding the consecutive pattern 1342.

%I A117156 #26 May 11 2019 10:15:41
%S A117156 1,1,2,6,23,110,630,4210,32150,276210,2636720,27687440,317169270,
%T A117156 3936056080,52603684760,753241509900,11504852242400,186705357825800,
%U A117156 3208160592252000,58188413286031600,1110946958902609400
%N A117156 Number of permutations avoiding the consecutive pattern 1342.
%C A117156 a(n) is the number of permutations on [n] that avoid the consecutive pattern 1342. It is the same as the number of permutations which avoid 2431, 4213, 3124, 1432, 2341, 4123 or 3214.
%D A117156 Sergi Elizalde, Asymptotic enumeration of permutations avoiding generalized patterns, Adv. Appl. Math. 36 (2006) 138-155.
%D A117156 Sergi Elizalde and Marc Noy, Consecutive patterns in permutations, Adv. Appl. Math. 30 (2003) 110-125.
%H A117156 Alois P. Heinz, <a href="/A117156/b117156.txt">Table of n, a(n) for n = 0..200</a> (terms n = 0..60 from Ray Chandler)
%H A117156 A. Baxter, B. Nakamura, and D. Zeilberger. <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/auto.html">Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes</a>
%H A117156 Steven Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/csolve/av.pdf">Pattern-Avoiding Permutations</a> [Broken link?]
%H A117156 Steven Finch, <a href="/A240885/a240885.pdf">Pattern-Avoiding Permutations</a> [Cached copy, with permission]
%F A117156 a(n) ~ c * d^n * n!, where d = 1/r = 0.9546118344740519430556804334164431663486451742931588346372174751881329..., where r = 1.04754620033697244977759528695194261... is the root of the equation integral_{x,0,r} exp(-x^3/6) dx = 1, and c = 1.1561985648406071020520797542907648300587978482957199521032311960968187467... . - _Vaclav Kotesovec_, Aug 23 2014
%t A117156 a[n_]:=Coefficient[Series[1/(1-Integrate[Exp[ -t^3/6],{t,0,x}]),{x,0,30}],x^n]*n!
%t A117156 (* Second program: *)
%t A117156 m = 21; gf = 1/(1-Sum[If[Mod[k, 3] == 0, (-1)^Mod[k, 6]/(6^(k/3)*(k/3)!), 0]* (x^(k+1)/(k+1)), {k, 0, m}]);
%t A117156 CoefficientList[gf + O[x]^m, x]*Range[0, m-1]! (* _Jean-François Alcover_, May 11 2019 *)
%Y A117156 Cf. A113228, A113229, A117158, A117226, A201692, A201693, A231166.
%Y A117156 Cf. A052324.
%K A117156 nonn
%O A117156 0,3
%A A117156 _Steven Finch_, Apr 26 2006