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A360975 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * A(x)^(3*n) / n!.

%I A360975 #10 Mar 13 2023 12:58:21
%S A360975 1,2,24,476,12380,386220,13821276,552876504,24318017424,1162989779660,
%T A360975 59987353249560,3316841570302680,195648523068917828,
%U A360975 12263065689662763024,814027053454694421000,57059908657536257254704,4212606733712173668180012,326799750176069289173027820
%N A360975 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * A(x)^(3*n) / n!.
%H A360975 Paul D. Hanna, <a href="/A360975/b360975.txt">Table of n, a(n) for n = 0..300</a>
%F A360975 G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
%F A360975 (1) A(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * A(x)^(3*n) / n!.
%F A360975 (2) A(x) = d/dx Series_Reversion(x - x^2*A(x)^3).
%F A360975 (3) B(x - x^2*A(x)^3) = x where B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * A(x)^(3*n) / n! ) is the g.f. of A360978.
%F A360975 (4) a(n) = (n+1) * A360978(n+1) for n >= 0.
%F A360975 a(n) ~ c * n! * n^alfa / LambertW(1/3)^n, where alfa = 2.598541481443... and c = 0.058191982295165... - _Vaclav Kotesovec_, Feb 28 2023
%F A360975 alfa = 7*LambertW(1/3) + 1/(1 + LambertW(1/3)). - _Vaclav Kotesovec_, Mar 13 2023
%e A360975 G.f.: A(x) = 1 + 2*x + 24*x^2 + 476*x^3 + 12380*x^4 + 386220*x^5 + 13821276*x^6 + 552876504*x^7 + 24318017424*x^8 + ... + a(n)*x^n + ...
%e A360975 where
%e A360975 A(x) = 1 + (d/dx x^2*A(x)^3) + (d^2/dx^2 x^4*A(x)^6)/2! + (d^3/dx^3 x^6*A(x)^9)/3! + (d^4/dx^4 x^8*A(x)^12)/4! + (d^5/dx^5 x^10*A(x)^15)/5! + (d^6/dx^6 x^12*A(x)^18)/6! + ... + (d^n/dx^n x^(2*n)*A(x)^(3*n))/n! + ...
%e A360975 Related series.
%e A360975 Let B(x) = Series_Reversion(x - x^2*A(x)^3), which begins
%e A360975 B(x) = x + x^2 + 8*x^3 + 119*x^4 + 2476*x^5 + 64370*x^6 + 1974468*x^7 + 69109563*x^8 + 2702001936*x^9 + ... + A360978(n)*x^n + ...
%e A360975 then A(x) = B'(x) and
%e A360975 B(x) = x * exp( x*A(x)^3 + (d/dx x^3*A(x)^6)/2! + (d^2/dx^2 x^5*A(x)^9)/3! + (d^3/dx^3 x^7*A(x)^12)/4! + (d^4/dx^4 x^9*A(x)^15)/5! + (d^5/dx^5 x^11*A(x)^18)/6! + ... + (d^(n-1)/dx^(n-1) x^(2*n-1)*A(x)^(3*n))/n! + ... ).
%o A360975 (PARI) {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
%o A360975 {a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, Dx(m, x^(2*m)*A^(3*m)/m!)) +O(x^(n+1))); polcoeff(A, n)}
%o A360975 for(n=0, 25, print1(a(n), ", "))
%o A360975 (PARI) /* Using series reversion (faster) */
%o A360975 {a(n) = my(A=1); for(i=1, n, A = deriv( serreverse(x - x^2*A^3 +O(x^(n+2))))); polcoeff(A, n)}
%o A360975 for(n=0, 25, print1(a(n), ", "))
%Y A360975 Cf. A360950, A360973, A360974, A360978.
%K A360975 nonn
%O A360975 0,2
%A A360975 _Paul D. Hanna_, Feb 27 2023