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A324130 Number of permutations of [n] that avoid the shuffle pattern s-k-t, where s = 1 and t = 132.

%I A324130 #26 Oct 31 2019 11:39:56
%S A324130 1,1,2,6,24,116,652,4178,30070,240164,2107606,20156458,208639514,
%T A324130 2323794632,27709659880,352203163790,4753474785808,67889631514128,
%U A324130 1022936113573148,16216615869916570,269816176058513398,4701111255106851632,85599799432794431978,1625828159969984754538
%N A324130 Number of permutations of [n] that avoid the shuffle pattern s-k-t, where s = 1 and t =  132.
%H A324130 Sergey Kitaev, <a href="http://dx.doi.org/10.1016/j.disc.2004.03.017">Partially Ordered Generalized Patterns</a>, Discrete Math. 298 (2005), no. 1-3, 212-229.
%F A324130 From _Petros Hadjicostas_, Oct 29 2019: (Start)
%F A324130 Let b(n) = A111004(n) = number of permutations avoiding a consecutive 132 pattern. Then a(n) = 2*a(n-1) - b(n-1) + Sum_{i = 1..n-1} binomial(n-1,i) * b(i) * a(n-1-i) for n >= 1 with a(0) = b(0) = 1. [See the recurrence for C_n on p. 220 of Kitaev (2005).]
%F A324130 E.g.f.: If A(x) is the e.g.f. of (a(n): n >= 0) and B(x) is the e.g.f. of (b(n): n >= 0) (i.e., B(x) = 1/(1 - Int(exp(-t^2/2), t = 0..x))), then A'(x) = (1 + B(x)) * A(x) - B(x) with A(0) = B(0) = 1. [Theorem 16, p. 219, in Kitaev (2005)] (End)
%Y A324130 Cf. A000142, A111004.
%K A324130 nonn
%O A324130 0,3
%A A324130 _N. J. A. Sloane_, Feb 16 2019
%E A324130 More terms from _Petros Hadjicostas_, Oct 29 2019 using a recurrence by Kitaev (2005)