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%I #24 Jan 04 2024 18:09:53
%S 1,1,3,15,102,870,8910,106470,1454040,22339800,381364200,7161323400,
%T 146701724400,3255661609200,77808668137200,1992415575150000,
%U 54420258228336000,1579320261543024000,48529229906613456000,1574046971727454224000,53741325186841612320000
%N a(n) is the number of rooted forests on n nodes that avoid the patterns 321, 2143, and 3142.
%C a(n) is the number of rooted labeled forests on n nodes so that along any path from the root to a vertex, there is at most one descent.
%H Andrew Howroyd, <a href="/A318618/b318618.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = n! + Sum_{k=1..n} Sum_{j=1..min(k, n-k)} (n!/2^j)*binomial(n-k-1, j-1)*binomial(k, j).
%t a[n_] := n! + Sum[n! 2^-j Binomial[n-k-1, j-1] Binomial[k, j], {k, 1, n}, {j, 1, Min[k, n-k]}];
%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Sep 13 2018 *)
%o (PARI) a(n) = {n! + sum(k=1, n, sum(j=1, min(k, n-k), n!/(2^j)*binomial(n-k-1,j-1)*binomial(k,j)))} \\ _Andrew Howroyd_, Aug 31 2018
%Y Cf. A000272, A318617, A007840, A000671, A000262.
%K nonn
%O 0,3
%A _Kassie Archer_, Aug 30 2018
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