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

1, 1, 2, 6, 24, 114, 608, 3554, 22480, 152546, 1103200, 8456994, 68411632, 581745250, 5183126016, 48245682338, 467988498064, 4720072211938, 49400302118560, 535546012710434, 6004045485933104, 69507152958422370, 829789019700511040, 10202854323325253538, 129061753086335478736

Offset

0,3

Comments

Stirling transform of j-> ceiling(2^(j-2)). - Alois P. Heinz, Aug 25 2021

Links

Alois P. Heinz, Table of n, a(n) for n = 0..558

Sergey Kitaev, Partially Ordered Generalized Patterns, Discrete Math. 298 (2005), no. 1-3, 212-229.

Formula

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

From Alois P. Heinz, Aug 25 2021: (Start)

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

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

Maple

b:= proc(n, m) option remember; `if`(n=0,
      ceil(2^(m-2)), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..24);  # Alois P. Heinz, Aug 25 2021

Mathematica

b[n_, m_] := b[n, m] = If[n == 0,
     Ceiling[2^(m-2)], m*b[n-1, m] + b[n-1, m+1]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Apr 15 2022, after Alois P. Heinz *)

Crossrefs

Cf. A000142, A001861, A035009, A347011.

Sequence in context: A209625 A054872 A134664 * A171448 A068199 A189846

Adjacent sequences: A324130 A324131 A324132 * A324134 A324135 A324136

Keyword

nonn

Author

N. J. A. Sloane, Feb 16 2019

Extensions

More terms from Petros Hadjicostas, Oct 30 2019

Status

approved