A234284 Number of 321-avoiding extensions of comb K_{s,3}^{beta}.

1, 10, 127, 1866, 29839, 504265, 8859742, 160216631, 2962451668, 55752953619, 1064455517286, 20566756704300, 401392396922394, 7901356125281267, 156695640175228660, 3127700524615849499, 62787047960901808378, 1266812106374162802049, 25675382888225888374354

Offset

1,2

Links

Table of n, a(n) for n=1..19.

C. Defant, Poset Pattern-Avoidance Problems Posed by Yakoubov, arXiv:1608.03951 [math.CO], 2016.

S. Yakoubov, Pattern Avoidance in Extensions of Comb-Like Posets, arXiv preprint arXiv:1310.2979 [math.CO], 2013.

Formula

Define F_{2,3}(k)=1 if 2<=k<=4 and 0 otherwise. For s>=3, let F_{s,3}(k) = Sum_{i=(s-1)..(k-1)} (F_{s-1,3}(i)*Sum_{j=(k-3s+4)..2} (binomial(k-i-1,j))). Then a(n) = Sum_{k=n..(3n-2)} (binomial(3n-k,2)*F_{n,3}(k)). - Colin Defant, Aug 16 2016

Mathematica

F[1, 3][1] = 1;
F[2, 3][k_] := If[2 <= k <= 4, 1, 0 ];
F[s_ /; s >= 3, 3][k_] := F[s, 3][k] = Sum[F[s - 1, 3][i] Sum[Binomial[k - i - 1, j], {j, k - 3s + 4, 2}], {i, s - 1, k - 1}];
a[n_] := Sum[Binomial[3n - k, 2] F[n, 3][k], {k, n, 3n - 2}];
Table[a[n], {n, 1, 19}] (* Jean-François Alcover, Jan 09 2019 *)

Crossrefs

Sequence in context: A270965 A245923 A218474 * A296379 A183538 A184678

Adjacent sequences: A234281 A234282 A234283 * A234285 A234286 A234287

Keyword

nonn

Author

N. J. A. Sloane, Dec 28 2013

Extensions

a(5)-a(9) from Colin Defant, Aug 16 2016

a(10)-a(19) from Alois P. Heinz, Aug 18 2016

Status

approved