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A360973 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A(x)^n / n!.

%I A360973 #12 Mar 13 2023 12:53:35
%S A360973 1,3,30,462,9243,223014,6232239,196780545,6905085381,266134485876,
%T A360973 11167349876424,506653305313320,24713399794830075,1289888647516181583,
%U A360973 71744797404818298768,4237233894492578488668,264870390088867267319274,17473793269024825938396135
%N A360973 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A(x)^n / n!.
%H A360973 Paul D. Hanna, <a href="/A360973/b360973.txt">Table of n, a(n) for n = 0..300</a>
%F A360973 G.f. A(x) = Sum_{n>=0} a(n)*x^(2*n) may be defined by the following.
%F A360973 (1) A(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A(x)^n / n!.
%F A360973 (2) A(x) = d/dx Series_Reversion(x - x^3*A(x)).
%F A360973 (3) B(x - x^3*A(x)) = x where B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(3*n-1) * A(x)^n / n! ) is the g.f. of A360976.
%F A360973 (4) a(n) = (2*n+1) * A360976(n+1) for n >= 0.
%F A360973 a(n) ~ c * n! * n^alfa * 2^n / LambertW(1)^n, where alfa = 1.50465021094584... and c = 0.36053267424501... - _Vaclav Kotesovec_, Feb 28 2023
%F A360973 alfa = (9*LambertW(1) - 1 + 3/(1 + LambertW(1)))/4. - _Vaclav Kotesovec_, Mar 13 2023
%e A360973 G.f.: A(x) = 1 + 3*x^2 + 30*x^4 + 462*x^6 + 9243*x^8 + 223014*x^10 + 6232239*x^12 + 196780545*x^14 + 6905085381*x^16 + ... + a(n)*x^(2*n) + ...
%e A360973 where
%e A360973 A(x) = 1 + (d/dx x^3*A(x)) + (d^2/dx^2 x^6*A(x)^2)/2! + (d^3/dx^3 x^9*A(x)^3)/3! + (d^4/dx^4 x^12*A(x)^4)/4! + (d^5/dx^5 x^15*A(x)^5)/5! + (d^6/dx^6 x^18*A(x)^6)/6! + ... + (d^n/dx^n x^(3*n)*A(x)^n)/n! + ...
%e A360973 Related series.
%e A360973 Let B(x) = Series_Reversion(x - x^3*A(x)), which begins
%e A360973 B(x) = x + x^3 + 6*x^5 + 66*x^7 + 1027*x^9 + 20274*x^11 + 479403*x^13 + 13118703*x^15 + 406181493*x^17 + ... + A360976(n)*x^(2*n-1) + ...
%e A360973 then A(x) = B'(x) and
%e A360973 B(x) = x * exp( x^2*A(x) + (d/dx x^5*A(x)^2)/2! + (d^2/dx^2 x^8*A(x)^3)/3! + (d^3/dx^3 x^11*A(x)^4)/4! + (d^4/dx^4 x^14*A(x)^5)/5! + (d^5/dx^5 x^17*A(x)^6)/6! + ... + (d^(n-1)/dx^(n-1) x^(3*n-1)*A(x)^n)/n! + ... ).
%o A360973 (PARI) {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
%o A360973 {a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, Dx(m, x^(3*m)*A^m/m!)) +O(x^(2*n+1))); polcoeff(A, 2*n)}
%o A360973 for(n=0, 25, print1(a(n), ", "))
%o A360973 (PARI) /* Using series reversion (faster) */
%o A360973 {a(n) = my(A=1); for(i=1, n, A = deriv( serreverse(x - x^3*A +O(x^(2*n+3))))); polcoeff(A, 2*n)}
%o A360973 for(n=0, 25, print1(a(n), ", "))
%Y A360973 Cf. A360950, A360974, A360975, A360976.
%Y A360973 Cf. A229619, A356848, A088714, A303063.
%K A360973 nonn
%O A360973 0,2
%A A360973 _Paul D. Hanna_, Feb 27 2023