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A274393 E.g.f. A(x) satisfies: A( A( x^3*exp(-3*x) )^(1/3) ) = x.

%I A274393 #14 Aug 12 2021 03:52:36
%S A274393 1,2,9,56,545,6696,100009,1756112,35480673,811332080,20696592521,
%T A274393 583009540488,17972297981521,601961695890296,21765379980020265,
%U A274393 844991974575946016,35056808550027808961,1547900555791042958688,72474037424646843142153,3586609339433026549298840,187062738581835989450074161,10255505482370456224398408872,589611389520200188085133153449
%N A274393 E.g.f. A(x) satisfies: A( A( x^3*exp(-3*x) )^(1/3) ) = x.
%H A274393 Vaclav Kotesovec, <a href="/A274393/b274393.txt">Table of n, a(n) for n = 1..175</a> (terms 1..100 from Paul D. Hanna)
%F A274393 E.g.f. A(x) = Sum_{n>=1} a(n) * x^n / n! satisfies:
%F A274393 (1) A( A( x^3*exp(3*x) )^(1/3) ) = -LambertW(-x*exp(x)).
%F A274393 (2) A(x) = Series_Reversion( A( x^3*exp(-3*x) )^(1/3) ).
%F A274393 (3) A( A(x)^3 * exp(-3*A(x)) ) = x^3.
%F A274393 a(n) ~ c * n! * d^n / n^(3/2), where d = 2.67119188675..., c = 0.3883596303... . - _Vaclav Kotesovec_, Aug 12 2021
%e A274393 E.g.f.: A(x) = x + 2*x^2/2! + 9*x^3/3! + 56*x^4/4! + 545*x^5/5! + 6696*x^6/6! + 100009*x^7/7! + 1756112*x^8/8! + 35480673*x^9/9! + 811332080*x^10/10! + 20696592521*x^11/11! + 583009540488*x^12/12! + 17972297981521*x^13/13! + 601961695890296*x^14/14! + 21765379980020265*x^15/15! +...
%e A274393 such that A( A( x^3*exp(-3*x) )^(1/3) ) = x.
%e A274393 RELATED SERIES.
%e A274393 The series reversion of the e.g.f. A(x) equals the series defined by:
%e A274393 A( x^3*exp(-3*x) )^(1/3) = x - 2*x^2/2! + 3*x^3/3! + 4*x^4/4! - 155*x^5/5! + 1914*x^6/6! - 15953*x^7/7! + 33592*x^8/8! + 2425257*x^9/9! - 71955530*x^10/10! + 1307665051*x^11/11! - 13707439692*x^12/12! - 125013414227*x^13/13! + 11742108426034*x^14/14! - 370418656051065*x^15/15! +...
%e A274393 Compare the above series reversion to the following series:
%e A274393 A(x)^3 * exp(-3*A(x)) = x^3 - 2*x^6/2! + 3*x^9/3! + 4*x^12/4! - 155*x^15/5! + 1914*x^18/6! - 15953*x^21/7! + 33592*x^24/8! +...
%e A274393 where A( A(x)^3 * exp(-3*A(x)) ) = x^3.
%e A274393 Note the following series is also in powers of x^3:
%e A274393 A(-A(x)^3 * exp(-3*A(x)) ) = -x^3 + 4*x^6/2! - 24*x^9/3! + 208*x^12/4! - 2400*x^15/5! + 36432*x^18/6! - 700672*x^21/7! + 16221088*x^24/8! - 434076480*x^27/9! + 13091390560*x^30/10! - 438602465664*x^33/11! + 16177344184080*x^36/12! - 652301794869088*x^39/13! + 28571154198355888*x^42/14! +...
%o A274393 (PARI) {a(n) = my(A=x); for(i=1, n, A = serreverse( subst(A, x, x^3*exp(-3*x +x*O(x^n)))^(1/3) ) ); n!*polcoeff(A, n)}
%o A274393 for(n=1, 30, print1(a(n), ", "))
%Y A274393 Cf. A274275, A274394, A274395.
%K A274393 nonn
%O A274393 1,2
%A A274393 _Paul D. Hanna_, Jun 21 2016