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%I #11 Apr 01 2016 14:20:15
%S 1,1,9,133,2729,71721,2300485,87194689,3815719969,189440927857,
%T 10525328121221,647265172064985,43660242639018241,3205987437435132793,
%U 254635755560090281525,21755037223870035810001,1989746853200670755116865,194000891136578173746676449,20089033883934411591428091013,2202022786357483714102765694185
%N E.g.f.: exp( T(T(T(T(x)))) ), where T(x) = -LambertW(-x) is Euler's tree function (A000169).
%H Vaclav Kotesovec, <a href="/A268654/b268654.txt">Table of n, a(n) for n = 0..100</a>
%F E.g.f. satisfies:
%F (1) A(x) = A(x/exp(x))^A(x).
%F (2) A(x) = exp( A(x)*T(T(T(x))) ).
%F (3) A(x/exp(x)) = exp(T(T(T(x)))) = LambertW(LambertW(LambertW(-x))) / LambertW(LambertW(-x)).
%F a(n) ~ exp(1 + (exp(-1) + exp(-1 - exp(-1)) + exp(-1 - exp(-1) - exp(-1 - exp(-1))))*n) * n^(n-1) / sqrt((1 - exp(-1)) * (1 + LambertW(LambertW(-exp(-1 - exp(-1) - exp(-1 - exp(-1)) - exp(-1 - exp(-1) - exp(-1 - exp(-1))))))) * (1 + LambertW(-exp(-1 - exp(-1) - exp(-1 - exp(-1)) - exp(-1 - exp(-1) - exp(-1 - exp(-1))))))). - _Vaclav Kotesovec_, Apr 01 2016
%e E.g.f.: A(x) = 1 + x + 9*x^2/2! + 133*x^3/3! + 2729*x^4/4! + 71721*x^5/5! + 2300485*x^6/6! + 87194689*x^7/7! + 3815719969*x^8/8! +...
%e where A(x) = A( x/exp(x) )^A(x).
%e RELATED SERIES.
%e Define W(x) = LambertW(-x)/(-x), where W(x) = exp(x*W(x)) and begins:
%e W(x) = 1 + x + 3*x^2/2! + 4^2*x^3/3! + 5^3*x^4/4! + 6^4*x^5/5! + 7^5*x^6/6! + 8^6*x^7/7! + 9^7*x^8/8! +...+ A000272(n+1)*x^n/n! +...
%e Let F(x) = A(x/exp(x)), which begins:
%e F(x) = 1 + x + 5*x^2/2! + 43*x^3/3! + 525*x^4/4! + 8321*x^5/5! + 162463*x^6/6! + 3774513*x^7/7! + 101808185*x^8/8! +...+ A227176(n)*x^n/n! +...
%e Let G(x) = F(x/exp(x)), which begins:
%e G(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! +...
%e then W(x), F(x), G(x), and A(x) are in the family of functions that begin:
%e (1) W(x) = exp(x)^W(x) = exp(T(x)),
%e (2) F(x) = W(x)^F(x) = exp(T(T(x))),
%e (3) G(x) = F(x)^G(x) = exp(T(T(T(x)))),
%e (4) A(x) = G(x)^A(x) = exp(T(T(T(T(x))))), ...
%e where T(x) = -LambertW(-x) is Euler's tree function:
%e T(x) = x + 2*x^2/2! + 3^2*x^3/3! + 4^3*x^4/4! + 5^4*x^5/5! + 6^5*x^6/! + 7^6*x^7/7! + 8^7*x^8/8! +...+ A000169(n)*x^n/n! +...
%t CoefficientList[Series[E^(-ProductLog[ProductLog[ProductLog[ProductLog[-x]]]]), {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Apr 01 2016 *)
%o (PARI) /* E.g.f.: A(x) = exp(T(T(T(T(x)))) ) */
%o {a(n)=local(T=sum(k=1, n, k^(k-1)*x^k/k!)+x*O(x^n)); n!*polcoeff(exp(subst(T, x, subst(T, x, subst(T, x, T)))), n)}
%o for(n=0, 25, print1(a(n), ", "))
%o (PARI) /* E.g.f.: A(x) = exp( -A(x)*LambertW(LambertW(LambertW(-x))) ) */
%o {a(n)=local(A=1+x, LambertW=sum(k=1, n, -k^(k-1)*(-x)^k/k!)+x*O(x^n));
%o for(i=1, n, A=exp(-A*subst(LambertW, x, subst(LambertW, x, subst(LambertW, x,-x))) +x*O(x^n))); n!*polcoeff(A, n)}
%o for(n=0, 25, print1(a(n), ", "))
%Y Cf. A268653, A227176, A000272, A000169.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Feb 18 2016
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