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A318617 a(n) is the number of rooted labeled forests on n nodes that avoid the patterns 213, 231, and 312.

%I A318617 #25 Jan 04 2024 18:09:57
%S A318617 1,1,3,13,73,503,4107,38773,415589,4986715,66238503,965102769,
%T A318617 15306905817,262567910999,4844199561787,95660129298709,
%U A318617 2013392566243565,44997370759528091,1064283567185090791,26560710262784693097,697529916604465424553
%N A318617 a(n) is the number of rooted labeled forests on n nodes that avoid the patterns 213, 231, and 312.
%H A318617 Andrew Howroyd, <a href="/A318617/b318617.txt">Table of n, a(n) for n = 0..200</a>
%H A318617 K. Anders and K. Archer, <a href="https://arxiv.org/abs/1607.03046">Rooted forests that avoid sets of permutations</a>, arXiv:1607.03046 [math.CO], 2017.
%F A318617 a(n) = Sum_{k=1..n} Sum_{r=1..k} binomial(n-1,k-1)*(r-1)!*a(n-k)*a(k-r) for n>0, a(0)=1.
%t A318617 a[0] = 1; a[n_] := a[n] = Sum[Binomial[n-1, k-1] (r-1)! a[n-k] a[k-r], {k, 1, n}, {r, 1, k}];
%t A318617 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Sep 13 2018 *)
%o A318617 (PARI) seq(n)={my(v=vector(n+1)); v[1]=1; for(n=1, n, v[n+1]=sum(k=1, n, sum(r=1, k, binomial(n-1,k-1)*(r-1)!*v[n-k+1]*v[k-r+1]))); v} \\ _Andrew Howroyd_, Aug 30 2018
%Y A318617 Cf. A000272, A318618, A007840, A000671, A000262.
%K A318617 nonn
%O A318617 0,3
%A A318617 _Kassie Archer_, Aug 30 2018