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%I #16 Dec 29 2014 19:40:16
%S 1,2,6,26,156,1242,12616,158034,2372880,41725106,843126624,
%T 19277549898,492447987136,13907344659210,430397513894016,
%U 14487404695687298,527023721684738304,20605894357093102434,861761850029367846400,38387125875316048363386,1814541564588778500135936
%N Number of ways to paint some (possibly none or all) of the trees over all forests on n labeled nodes.
%H Alois P. Heinz, <a href="/A218691/b218691.txt">Table of n, a(n) for n = 0..150</a>
%F E.g.f.: exp(T(x) - T(x)^2/2)^2 where T(x) is e.g.f. for A000169.
%F a(n) = Sum_{m=1..n} A105599(n,m)*2^m.
%F a(n) ~ 2*n^(n-2)*exp(1). - _Vaclav Kotesovec_, Aug 16 2013
%p T:= -LambertW(-x):
%p egf:= exp(T-T^2/2)^2:
%p a:= n-> n! * coeff(series(egf, x, n+1), x, n):
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Nov 04 2012
%t nn=20;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Range[0,nn]!CoefficientList[ Series[Exp[(t-t^2/2)]^2,{x,0,nn}],x]
%Y Cf. A101313.
%K nonn
%O 0,2
%A _Geoffrey Critzer_, Nov 04 2012
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