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%I #71 May 20 2023 23:16:51
%S 1,1,2,6,23,110,630,4204,32054,274914,2619692,27459344,313990182,
%T 3889585408,51888955808,741668212080,11307669002720,183174676857608,
%U 3141820432768752,56882461258572976,1084056190235653304,21692744773505849952,454758269790599361968
%N Number of permutations of [n] avoiding the consecutive pattern 1243.
%C a(n) is the number of permutations on [n] that avoid the consecutive pattern 1243. It is the same as the number of permutations which avoid 3421, 4312 or 2134.
%H Alois P. Heinz, <a href="/A117226/b117226.txt">Table of n, a(n) for n = 0..200</a> (terms n = 0..60 from Ray Chandler)
%H 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>, 2011.
%H Sergi Elizalde and Marc Noy, <a href="https://doi.org/10.1016/S0196-8858(02)00527-4 ">Consecutive patterns in permutations</a>, Adv. Appl. Math. 30 (2003), 110-125; see p. 120.
%H Sergi Elizalde, <a href="https://arxiv.org/abs/math/0505254">Asymptotic enumeration of permutations avoiding generalized patterns</a>, arXiv:math/0505254 [math.CO], 2005.
%H Sergi Elizalde, <a href="https://doi.org/10.1016/j.aam.2005.05.006">Asymptotic enumeration of permutations avoiding generalized patterns</a>, Adv. Appl. Math. 36 (2006), 138-155.
%H Steven Finch, <a href="https://web.archive.org/web/20150405060634/http://www.people.fas.harvard.edu/~sfinch/csolve/av.pdf">Pattern-Avoiding Permutations</a>. [Archived copy]
%H Steven Finch, <a href="/A240885/a240885.pdf">Pattern-Avoiding Permutations</a>. [Cached copy, with permission]
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PochhammerSymbol.html">Pochhammer Symbol</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Falling_and_rising_factorials">Falling and rising factorials</a>.
%F a(n) ~ c * d^n * n!, where d = 0.952891423325053197208702817349165942637814..., c = 1.169657787464830219717093446929792145316... . - _Vaclav Kotesovec_, Aug 23 2014
%F From _Petros Hadjicostas_, Nov 01 2019:
%F E.g.f.: 1/W(z), where W(z) := 1 + Sum_{n >= 0} (-1)^(n+1)* z^(3*n+1)/(b(n)*(3*n+1)) with b(n) = A176730(n) = (3*n)!/(3^n*(1/3)_n). (Here (x)_n = x*(x + 1)*...*(x + n - 1) is the Pochhammer symbol, or rising factorial, which is denoted by (x)^n in some papers and books.) The function W(z) satisfies the o.d.e. W'''(z) + z*W'(z) = 0 with W(0) = 1, W'(0) = -1, and W''(0) = 0. [See Theorem 4.3 (Case 1243 with u = 0) in Elizalde and Noy (2003).]
%F a(n) = Sum_{m = 0..floor((n-1)/3)} (-3)^m * (1/3)_m * binomial(n, 3*m+1) * a(n-3*m-1) for n >= 1 with a(0) = 1. (End)
%p b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
%p add(b(u-j, o+j-1, 0), j=`if`(t<0, -t, 1)..u)+
%p add(b(u+j-1, o-j, `if`(t=0, j, -j)), j=1..o))
%p end:
%p a:= n-> b(n, 0$2):
%p seq(a(n), n=0..25); # _Alois P. Heinz_, Nov 07 2013
%t A[x_]:=Integrate[AiryAi[ -t],{t,0,x}]; B[x_]:=Integrate[AiryBi[ -t],{t,0,x}];
%t c=-3^(2/3)*Gamma[2/3]/2; d=-3^(1/6)*Gamma[2/3]/2;
%t a[n_]:=SeriesCoefficient[1/(c*A[x]+d*B[x]+1),{x,0,n}]*n!; Table[a[n],{n,0,10}] (* fixed by _Vaclav Kotesovec_, Aug 23 2014 *)
%t (* constant d: *) 1/x/.FindRoot[3^(2/3)*Gamma[2/3]/2 * Integrate[AiryAi[-t],{t,0,x}] + 3^(1/6)*Gamma[2/3]/2 * Integrate[AiryBi[-t],{t,0,x}]==1,{x,1},WorkingPrecision->50] (* _Vaclav Kotesovec_, Aug 23 2014 *)
%Y Cf. A113228, A113229, A117156, A117158, A176730, A201692, A201693, A231166.
%Y Row m = 1 of A327722.
%K nonn
%O 0,3
%A _Steven Finch_, Apr 26 2006
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