A368636 Number of modified ascent sequences of length n avoiding the pattern 221.

1, 1, 2, 5, 14, 44, 155, 607, 2617, 12306, 62587, 341790, 1991916, 12324031, 80587935, 554826429, 4008364544, 30299290911, 239019427636, 1963239741712, 16755637216417, 148317595764043, 1359380603278377, 12880841117125364, 126007744452786277, 1270998629233371388

Offset

0,3

Links

Table of n, a(n) for n=0..25.

Giulio Cerbai, Pattern-avoiding modified ascent sequences, arXiv:2401.10027 [math.CO], 2024.

Formula

a(n) = Sum_{k=1..n} Sum_{i=1..k} S2(k-1,i-1) * binomial(n-1-k+i,i-1) for n >= 1, a(0)=1, where S2(n,i) are the Stirling numbers of the second kind.

Example

The shortest modified ascent sequence that contains 221 is 1221.

Mathematica

a[0]=1; a[n_]:=Sum[Sum[StirlingS2[k-1, i-1] Binomial[n-1-k+i, i-1], {i, k}], {k, n}]; Array[a, 26, 0] (* Stefano Spezia, Jan 20 2024 *)

Crossrefs

Cf. A022493 (all modified ascents).

Sequence in context: A350492 A014322 A095148 * A060996 A134378 A101226

Adjacent sequences: A368633 A368634 A368635 * A368637 A368638 A368639

Keyword

nonn

Author

Giulio Cerbai, Jan 19 2024

Status

approved