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%I #9 Oct 04 2020 11:42:48
%S 1,1,3,22,329,8636,355297,21117286,1710243761,180811765432,
%T 24158025584801,3977274470362634,790696358461658761,
%U 186695449895152470052,51635196859642278380513,16532803795918313120452246
%N E.g.f.: A(x) = exp(x*exp(x*exp(2*x*exp(3*x*exp(...exp(n*x*exp(...))...))))).
%H Vaclav Kotesovec, <a href="/A189897/b189897.txt">Table of n, a(n) for n = 0..242</a>
%F E.g.f.: A(x) = exp(x*B(x)) where B(x) is the e.g.f. of A096537.
%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 22*x^3/3! + 329*x^4/4! + 8636*x^5/5! +...
%e The e.g.f. and related series satisfy:
%e A(x) = exp(x*B), B = exp(x*C^2), C = exp(x*D^3), D = exp(x*E^4), E = exp(x*F^5), F = exp(x*G^6), ...
%e where the series begin:
%e B = 1 + x + 5*x^2/2! + 61*x^3/3! + 1377*x^4/4! + 49721*x^5/5! +...
%e C = 1 + x + 7*x^2/2! + 118*x^3/3! + 3529*x^4/4! + 162076*x^5/5! +...
%e D = 1 + x + 9*x^2/2! + 193*x^3/3! + 7169*x^4/4! + 399521*x^5/5! +...
%e E = 1 + x + 11*x^2/2! + 286*x^3/3! + 12681*x^4/4! + 830876*x^5/5! +...
%e F = 1 + x + 13*x^2/2! + 397*x^3/3! + 20449*x^4/4! + 1539961*x^5/5! +...
%e G = 1 + x + 15*x^2/2! + 526*x^3/3! + 30857*x^4/4! + 2625596*x^5/5! +...
%e Relevant powers of the above series begin:
%e C^2 = 1 + 2*x + 16*x^2/2! + 278*x^3/3! + 8296*x^4/4! + 375962*x^5/5! +...
%e D^3 = 1 + 3*x + 33*x^2/2! + 747*x^3/3! + 27921*x^4/4! + 1536723*x^5/5! +...
%e E^4 = 1 + 4*x + 56*x^2/2! + 1564*x^3/3! + 70416*x^4/4! + 4576724*x^5/5! +...
%e F^5 = 1 + 5*x + 85*x^2/2! + 2825*x^3/3! + 148945*x^4/4! + 11182925*x^5/5! +...
%e G^6 = 1 + 6*x + 120*x^2/2! + 4626*x^3/3! + 279672*x^4/4! + 23840046*x^5/5! +...
%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(x*(n-i+1)*A+x*O(x^n))); n!*polcoeff(exp(x*A), n)}
%Y Cf. A096537.
%K nonn
%O 0,3
%A _Paul D. Hanna_, May 01 2011
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