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
Keywords
Examples
The shortest modified ascent sequence that contains 221 is 1221.
Links
- Giulio Cerbai, Pattern-avoiding modified ascent sequences, arXiv:2401.10027 [math.CO], 2024.
Crossrefs
Cf. A022493 (all modified ascents).
Programs
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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 *)
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.