A324133 - OEIS

%I #34 Apr 15 2022 10:32:14

%S 1,1,2,6,24,114,608,3554,22480,152546,1103200,8456994,68411632,

%T 581745250,5183126016,48245682338,467988498064,4720072211938,

%U 49400302118560,535546012710434,6004045485933104,69507152958422370,829789019700511040,10202854323325253538,129061753086335478736

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

%C Stirling transform of j-> ceiling(2^(j-2)). - _Alois P. Heinz_, Aug 25 2021

%H Alois P. Heinz, <a href="/A324133/b324133.txt">Table of n, a(n) for n = 0..558</a>

%H 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 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 From _Alois P. Heinz_, Aug 25 2021: (Start)

%F G.f.: Sum_{k>=0} ceiling(2^(k-2))*x^k / Product_{j=1..k} (1-j*x).

%F a(n) = Sum_{j=0..n} Stirling2(n,j)*ceiling(2^(j-2)). (End)

%p b:= proc(n, m) option remember; `if`(n=0,

%p       ceil(2^(m-2)), m*b(n-1, m)+b(n-1, m+1))

%p     end:

%p a:= n-> b(n, 0):

%p seq(a(n), n=0..24);  # _Alois P. Heinz_, Aug 25 2021

%t b[n_, m_] := b[n, m] = If[n == 0,

%t      Ceiling[2^(m-2)], m*b[n-1, m] + b[n-1, m+1]];

%t a[n_] := b[n, 0];

%t Table[a[n], {n, 0, 24}] (* _Jean-François Alcover_, Apr 15 2022, after _Alois P. Heinz_ *)

%Y Cf. A000142, A001861, A035009, A347011.

%K nonn

%O 0,3

%A _N. J. A. Sloane_, Feb 16 2019

%E More terms from _Petros Hadjicostas_, Oct 30 2019