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A229619 G.f. satisfies: A(x) = Series_Reversion(x - x^2*A'(x)).

%I A229619 #24 Feb 24 2025 06:26:28
%S A229619 1,1,4,27,248,2822,37820,578915,9918924,187558638,3873705128,
%T A229619 86692262942,2089070253556,53925007946392,1484529898970648,
%U A229619 43421639185592359,1344923240469786704,43981996770022295714,1514531024603022580980,54783958839510354056018,2077007174758224026216216
%N A229619 G.f. satisfies: A(x) = Series_Reversion(x - x^2*A'(x)).
%C A229619 a(n) = A360950(n-1)/n for n >= 1. [corrected by _Vaclav Kotesovec_, Feb 27 2023]
%H A229619 Vaclav Kotesovec, <a href="/A229619/b229619.txt">Table of n, a(n) for n = 1..300</a>
%F A229619 G.f. satisfies: A(x) = x + A(x)^2 * A'(A(x)).
%F A229619 a(n) ~ c * n! * n^(3*LambertW(1) - 2 + 1/(1 + LambertW(1))) / LambertW(1)^n, where c = 0.109236306585641816289... - _Vaclav Kotesovec_, Feb 27 2023
%e A229619 G.f.: A(x) = x + x^2 + 4*x^3 + 27*x^4 + 248*x^5 + 2822*x^6 + ...
%e A229619 By definition, A(x - x^2*A'(x)) = x, where
%e A229619 A'(x) = 1 + 2*x + 12*x^2 + 108*x^3 + 1240*x^4 + 16932*x^5 + 264740*x^6 + 4631320*x^7 + ... + A360950(n)*x^n + ...
%e A229619 Related expansions.
%e A229619 A'(A(x)) = 1 + 2*x + 14*x^2 + 140*x^3 + 1726*x^4 + 24752*x^5 + ...
%e A229619 A(x)^2 = x^2 + 2*x^3 + 9*x^4 + 62*x^5 + 566*x^6 + 6356*x^7 + ...
%o A229619 (PARI) {a(n)=local(A=x+x^2);for(i=1,n,A=serreverse(x-x^2*A'+x*O(x^n)));polcoeff(A,n)}
%o A229619 for(n=1,25,print1(a(n),", "))
%Y A229619 Cf. A360950.
%K A229619 nonn
%O A229619 1,3
%A A229619 _Paul D. Hanna_, Sep 26 2013