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A324133 Number of permutations of [n] that avoid the shuffle pattern s-k-t, where s = 12 and t = 12.

%I A324133 #34 Apr 15 2022 10:32:14
%S A324133 1,1,2,6,24,114,608,3554,22480,152546,1103200,8456994,68411632,
%T A324133 581745250,5183126016,48245682338,467988498064,4720072211938,
%U A324133 49400302118560,535546012710434,6004045485933104,69507152958422370,829789019700511040,10202854323325253538,129061753086335478736
%N A324133 Number of permutations of [n] that avoid the shuffle pattern s-k-t, where s = 12 and t = 12.
%C A324133 Stirling transform of j-> ceiling(2^(j-2)). - _Alois P. Heinz_, Aug 25 2021
%H A324133 Alois P. Heinz, <a href="/A324133/b324133.txt">Table of n, a(n) for n = 0..558</a>
%H A324133 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 A324133 a(n) = -2^(n-1) + 2*Sum_{i = 0..n-1} binomial(n-1,i) * a(i) with a(0) = 1. [It follows from Kitaev's recurrence for C_n on p. 220 of his paper.] - _Petros Hadjicostas_, Oct 30 2019
%F A324133 From _Alois P. Heinz_, Aug 25 2021: (Start)
%F A324133 G.f.: Sum_{k>=0} ceiling(2^(k-2))*x^k / Product_{j=1..k} (1-j*x).
%F A324133 a(n) = Sum_{j=0..n} Stirling2(n,j)*ceiling(2^(j-2)). (End)
%p A324133 b:= proc(n, m) option remember; `if`(n=0,
%p A324133       ceil(2^(m-2)), m*b(n-1, m)+b(n-1, m+1))
%p A324133     end:
%p A324133 a:= n-> b(n, 0):
%p A324133 seq(a(n), n=0..24);  # _Alois P. Heinz_, Aug 25 2021
%t A324133 b[n_, m_] := b[n, m] = If[n == 0,
%t A324133      Ceiling[2^(m-2)], m*b[n-1, m] + b[n-1, m+1]];
%t A324133 a[n_] := b[n, 0];
%t A324133 Table[a[n], {n, 0, 24}] (* _Jean-François Alcover_, Apr 15 2022, after _Alois P. Heinz_ *)
%Y A324133 Cf. A000142, A001861, A035009, A347011.
%K A324133 nonn
%O A324133 0,3
%A A324133 _N. J. A. Sloane_, Feb 16 2019
%E A324133 More terms from _Petros Hadjicostas_, Oct 30 2019