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A360974 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * A(x)^(2*n) / n!.

%I A360974 #11 Mar 13 2023 12:55:56
%S A360974 1,2,18,260,4890,110124,2844772,82196424,2613699450,90450874860,
%T A360974 3379153837180,135445714293720,5796441493971284,263784018974675416,
%U A360974 12721572505160772840,648250134428292640272,34809708051186914034730,1965040180185473309749788,116359823755204505172646204
%N A360974 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * A(x)^(2*n) / n!.
%H A360974 Paul D. Hanna, <a href="/A360974/b360974.txt">Table of n, a(n) for n = 0..299</a>
%F A360974 G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
%F A360974 (1) A(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * A(x)^(2*n) / n!.
%F A360974 (2) A(x) = d/dx Series_Reversion(x - x^2*A(x)^2).
%F A360974 (3) B(x - x^2*A(x)^2) = x where B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * A(x)^(2*n) / n! ) is the g.f. of A360977.
%F A360974 (4) a(n) = (n+1) * A360977(n+1) for n >= 0.
%F A360974 a(n) ~ c * n! * n^alfa / LambertW(1/2)^n, where alfa = 2.498459235192... and c = 0.0920029178453... - _Vaclav Kotesovec_, Feb 28 2023
%F A360974 alfa = 5*LambertW(1/2) + 1/(1 + LambertW(1/2)). - _Vaclav Kotesovec_, Mar 13 2023
%e A360974 G.f.: A(x) = 1 + 2*x + 18*x^2 + 260*x^3 + 4890*x^4 + 110124*x^5 + 2844772*x^6 + 82196424*x^7 + 2613699450*x^8 + ... + a(n)*x^n + ...
%e A360974 where
%e A360974 A(x) = 1 + (d/dx x^2*A(x)^2) + (d^2/dx^2 x^4*A(x)^4)/2! + (d^3/dx^3 x^6*A(x)^6)/3! + (d^4/dx^4 x^8*A(x)^8)/4! + (d^5/dx^5 x^10*A(x)^10)/5! + (d^6/dx^6 x^12*A(x)^12)/6! + ... + (d^n/dx^n x^(2*n)*A(x)^(2*n))/n! + ...
%e A360974 Related series.
%e A360974 Let B(x) = Series_Reversion(x - x^2*A(x)^2), which begins
%e A360974 B(x) = x + x^2 + 6*x^3 + 65*x^4 + 978*x^5 + 18354*x^6 + 406396*x^7 + 10274553*x^8 + 290411050*x^9 + ... + A360977(n)*x^n + ...
%e A360974 then A(x) = B'(x) and
%e A360974 B(x) = x * exp( x*A(x)^2 + (d/dx x^3*A(x)^4)/2! + (d^2/dx^2 x^5*A(x)^6)/3! + (d^3/dx^3 x^7*A(x)^8)/4! + (d^4/dx^4 x^9*A(x)^10)/5! + (d^5/dx^5 x^11*A(x)^12)/6! + ... + (d^(n-1)/dx^(n-1) x^(2*n-1)*A(x)^(2*n))/n! + ... ).
%o A360974 (PARI) {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
%o A360974 {a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, Dx(m, x^(2*m)*A^(2*m)/m!)) +O(x^(n+1))); polcoeff(A, n)}
%o A360974 for(n=0, 25, print1(a(n), ", "))
%o A360974 (PARI) /* Using series reversion (faster) */
%o A360974 {a(n) = my(A=1); for(i=1, n, A = deriv( serreverse(x - x^2*A^2 +O(x^(n+2))))); polcoeff(A, n)}
%o A360974 for(n=0, 25, print1(a(n), ", "))
%Y A360974 Cf. A360950, A360973, A360975, A360977.
%Y A360974 Cf. A229619, A356848, A088714, A303063.
%K A360974 nonn
%O A360974 0,2
%A A360974 _Paul D. Hanna_, Feb 27 2023