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%I #29 Aug 04 2019 19:29:44
%S 1,4,30,332,4880,89742,1986124,51471800,1530489744,51395228090,
%T 1924687118684,79553145323940,3598161485778808,176797212122233094,
%U 9378715234039802340,534259395682874552048,32528761111972930621472,2108146039402630977388530,144899759883703796130871468,10528261771566724089621962780
%N E.g.f.: T(T(x)), where T(x) is the e.g.f. for labeled rooted trees, A000169.
%C Exponential series reversal gives A185298 with alternating signs: 1, -4, 18, -92, 520, ... . - _Vladimir Reshetnikov_, Aug 04 2019
%H Vincenzo Librandi, <a href="/A207833/b207833.txt">Table of n, a(n) for n = 1..200</a>
%H Yoshida Tomoyuki, <a href="http://dx.doi.org/10.1006/jabr.2000.8602">Categorical aspects of generating functions. I. Exponential formulas and Krull-Schmidt categories</a>, J. Algebra 240 (2001), no. 1, 40-82. MR1830543 (2002e:18008). See Sect. 6.8.
%F a(n) = 1/n * Sum_{k=1..n} C(n,k)*k^k*n^(n-k). [_Vladimir Kruchinin_, Sep 24 2012]
%F a(n) = n^(n-1) - Sum_{k=1..n-1} (-1)^(n-k) * C(n, k) * k^(n-k) * a(k) for n>1 with a(1)=1. - _Paul D. Hanna_, Nov 21 2012
%F E.g.f. A(x) satisfies: A(x) = Sum_{n>=1} n^(n-1)*T(x)^n/n!, by definition.
%F E.g.f. A(x) satisfies: A(x/exp(x)) = T(x) = Sum_{n>=1} n^(n-1)*x^n/n!. - _Paul D. Hanna_, Jul 04 2013
%F a(n) ~ n^(n-1) * exp(n*exp(-1)) / sqrt(1-exp(-1)). - _Vaclav Kotesovec_, Feb 24 2014
%e E.g.f.: A(x) = x + 4*x^2/2! + 30*x^3/3! + 332*x^4/4! + 4880*x^5/5! +...
%e Euler's tree function T(x) satisfies: T(x/exp(x)) = x, and begins:
%e T(x) = x + 2*x^2/2! + 3^2*x^3/3! + 4^3*x^4/4! + 5^4*x^5/5! +...+ A000169(n)*x^n/n! +...
%e where e.g.f. A(x) = T(T(x)).
%t nn=20;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Range[ 0,nn]!CoefficientList[ ComposeSeries[ Series[t,{x,0,nn}],Series[t,{x,0,nn}]],x] (* _Geoffrey Critzer_, Sep 16 2012 *)
%t Rest[CoefficientList[Series[-LambertW[LambertW[-x]], {x, 0, 20}], x] * Range[0, 20]!] (* _Vaclav Kotesovec_, Feb 24 2014 *)
%o (PARI) {a(n)=if(n==0|n==1, 1, n^(n-1)-sum(k=1, n-1, (-1)^(n-k)*binomial(n, k)*k^(n-k)*a(k)))} \\ _Paul D. Hanna_, Nov 21 2012
%Y Cf. A000169, A185298, A227278, A227176.
%K nonn
%O 1,2
%A _N. J. A. Sloane_, Feb 20 2012
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