A234284 - OEIS

%I #20 Jan 09 2019 09:11:55

%S 1,10,127,1866,29839,504265,8859742,160216631,2962451668,55752953619,

%T 1064455517286,20566756704300,401392396922394,7901356125281267,

%U 156695640175228660,3127700524615849499,62787047960901808378,1266812106374162802049,25675382888225888374354

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

%H C. Defant, <a href="http://arxiv.org/abs/1608.03951">Poset Pattern-Avoidance Problems Posed by Yakoubov</a>, arXiv:1608.03951 [math.CO], 2016.

%H S. Yakoubov, <a href="http://arxiv.org/abs/1310.2979">Pattern Avoidance in Extensions of Comb-Like Posets</a>, arXiv preprint arXiv:1310.2979 [math.CO], 2013.

%F 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

%t F[1, 3][1] = 1;

%t F[2, 3][k_] := If[2 <= k <= 4, 1, 0 ];

%t 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}];

%t a[n_] := Sum[Binomial[3n - k, 2] F[n, 3][k], {k, n, 3n - 2}];

%t Table[a[n], {n, 1, 19}] (* _Jean-François Alcover_, Jan 09 2019 *)

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Dec 28 2013

%E a(5)-a(9) from _Colin Defant_, Aug 16 2016

%E a(10)-a(19) from _Alois P. Heinz_, Aug 18 2016