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Number of forests of labeled rooted trees of height at most 1, with n labels, where each root contains a nonempty set of labels of equal size, also row sums of A143398.
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%I #16 Mar 29 2016 20:44:13

%S 1,1,4,14,55,252,1319,7737,50040,351636,2659375,21519027,185279186,

%T 1688183135,16206401020,163376811610,1724624368377,19011582728772,

%U 218312877627483,2605840967052663,32271957793959066,413991491885677105,5492584623675060620

%N Number of forests of labeled rooted trees of height at most 1, with n labels, where each root contains a nonempty set of labels of equal size, also row sums of A143398.

%H Alois P. Heinz, <a href="/A143406/b143406.txt">Table of n, a(n) for n = 0..530</a>

%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>

%F a(n) = 1 if n=0 and a(n) = n! * Sum_{k=1..n} Sum_{i=1..floor(n/k)} i^(n-k*i)/ ((n-k*i)!*i!*k!^i) else.

%e a(2) = 4, because 4 forests with 2 labels exist: {1}{2}, {1}<-2, {2}<-1, {1,2}.

%p a:= n-> if n=0 then 1 else n! * add(add(i^(n-k*i)/

%p ((n-k*i)!*i!*k!^i), i=1..floor(n/k)), k=1..n) fi:

%p seq(a(n), n=0..30);

%Y Cf. A143398, A000142.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Aug 12 2008