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%I #24 Nov 11 2019 00:30:51
%S 1,3,10,77,626,8707,117650,2242193,43250842,1049248991,25937424602,
%T 772559330281,23298085122482,817466439388341,29223801257127976,
%U 1181267018656911617,48661191875666868482,2232302772999145783735,104127350297911241532842
%N Number of forests on n vertices consisting of labeled rooted trees of the same size.
%H Alois P. Heinz, <a href="/A236696/b236696.txt">Table of n, a(n) for n = 1..200</a>
%F a(n) = sum(d divides n, n!/(n/d)!*(d^(d-1)/d!)^(n/d) ).
%F E.g.f.: sum(k>=1, exp(k^(k-1)*x^k/k!)).
%e For n = 3 we have the following 10 forests (where the roots are denoted by ^):
%e 3 2 3 1 2 1
%e | | | | | |
%e 2 3 1 3 1 2 2 3 1 3 1 2
%e \ / \ / \ / | | | | | |
%e 1 2 3 1 2 3 1 1 2 2 3 3
%e ^ ^ ^, ^, ^, ^, ^, ^, ^, ^, ^, ^
%t Table[Sum[n!/(n/d)!*(d^(d-1)/d!)^(n/d), {d,Divisors[n]}], {n,1,100}]
%o (Maxima) a(n):= lsum(n!/(n/d)!*(d^(d-1)/d!)^(n/d),d,listify(divisors(n))); makelist(a(n),n,1,40); /* _Emanuele Munarini_, Feb 03 2014 */
%Y Cf. A000169, A038041, A005225, A055225.
%K nonn
%O 1,2
%A _Emanuele Munarini_, Jan 30 2014
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