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%I #22 Jan 26 2020 16:36:48
%S 1,1,3,15,106,975,11106,151501,2415960,44221869,915826600,21211128411,
%T 544126606992,15334985416075,471495297242256,15719617534811625,
%U 565271886957356416,21820620411482896089,900398349688515500160,39564926462522623540519,1845034125763359894240000
%N Number of ways to linearly arrange the trees over all forests on n labeled nodes.
%H Alois P. Heinz, <a href="/A218688/b218688.txt">Table of n, a(n) for n = 0..150</a>
%F E.g.f.: 1/(1- T(x) + T(x)^2/2) where T(x) is e.g.f. for A000169.
%F a(n) = Sum_{m=1..n} A105599(n,m)*m!.
%F a(n) ~ 4*n^(n-2). - _Vaclav Kotesovec_, Aug 16 2013
%F a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * k^(k-2) * a(n-k). - _Ilya Gutkovskiy_, Jan 26 2020
%p T:= -LambertW(-x):
%p egf:= 1/(1-T+T^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[1/(1-t+t^2/2),{x,0,nn}],x]
%o (PARI) A218688_vec(n,A=List(1))={until(#A>n,listput(A,sum(k=1,#A,binomial(#A,k)*k^(k-2)*A[#A-k+1])));Vec(A)} \\ _M. F. Hasler_, Jan 26 2020
%Y Cf. A101313.
%K nonn
%O 0,3
%A _Geoffrey Critzer_, Nov 04 2012
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