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A227278 E.g.f.: T(T(T(x))), where T(x) = -LambertW(-x) is Euler's tree function (A000169).

%I A227278 #14 Feb 11 2016 12:37:50
%S A227278 1,6,63,948,18645,454158,13221075,448434136,17386204761,759123121050,
%T A227278 36882981687519,1974616464026484,115536647641839333,
%U A227278 7336947898087080406,502660682907018997755,36961205206337621142192,2903732354672613314658225,242753209611983811853905330
%N A227278 E.g.f.: T(T(T(x))), where T(x) = -LambertW(-x) is Euler's tree function (A000169).
%H A227278 Alois P. Heinz, <a href="/A227278/b227278.txt">Table of n, a(n) for n = 1..100</a>
%F A227278 Given e.g.f. A(x), A(x/exp(x)) = A(x)/exp(A(x)) = T(T(x)) and equals the e.g.f. of A207833.
%F A227278 a(n) ~ n! * exp((1+exp(-1)+exp(-1-exp(-1)))*n)/(sqrt(2*Pi*(1-exp(-1))*(1-exp(-1-exp(-1))))*n^(3/2)). - _Vaclav Kotesovec_, Jul 05 2013
%e A227278 E.g.f.: A(x) = x + 6*x^2/2! + 63*x^3/3! + 948*x^4/4! + 18645*x^5/5! +...
%e A227278 Euler's tree function T(x) satisfies: T(x/exp(x)) = x, and begins:
%e A227278 T(x) = x + 2*x^2/2! + 3^2*x^3/3! + 4^3*x^4/4! + 5^4*x^5/5! +...
%e A227278 where A(x) = T(T(T(x))).
%e A227278 Related expansions:
%e A227278 A(x/exp(x)) = A(x)/exp(A(x)) = x + 4*x^2/2! + 30*x^3/3! + 332*x^4/4! + 4880*x^5/5! + 89742*x^6/6! + 1986124*x^7/7! + 51471800*x^8/8! +...+ A207833(n)*x^n/n! +...
%e A227278 exp(A(x)) = 1 + x + 7*x^2/2! + 82*x^3/3! + 1345*x^4/4! + 28396*x^5/5! + 734149*x^6/6! + 22485898*x^7/7! + 796769201*x^8/8! +...+ A268653(n)*x^n/n! +...
%t A227278 Rest[CoefficientList[Series[-LambertW[LambertW[LambertW[-x]]], {x, 0, 20}], x]* Range[0, 20]!] (* _Vaclav Kotesovec_, Jul 05 2013 *)
%o A227278 (PARI) /* E.g.f.: A(x) = T(T(T(x))) */
%o A227278 {a(n)=local(T=sum(k=1, n, k^(k-1)*x^k/k!)+x*O(x^n)); n!*polcoeff(subst(T,x,subst(T, x, T)), n)}
%o A227278 for(n=1, 20, print1(a(n), ", "))
%o A227278 (PARI) /* E.g.f.: A(x) = -LambertW(LambertW(LambertW(-x))) */
%o A227278 {a(n)=local(LambertW=sum(k=1, n, -k^(k-1)*(-x)^k/k!)+x*O(x^n));
%o A227278 n!*polcoeff(-subst(LambertW,x,subst(LambertW,x,subst(LambertW,x,-x))),n)}
%o A227278 for(n=1, 20, print1(a(n), ", "))
%Y A227278 Cf. A268653, A207833, A000169.
%K A227278 nonn
%O A227278 1,2
%A A227278 _Paul D. Hanna_, Jul 04 2013