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A361049 G.f. satisfies: A(x) = (1/x)*Series_Reversion( x/(1 + x*A(x)^2 + x^2*A(x)*A'(x)) ).

%I A361049 #7 Mar 13 2023 12:47:24
%S A361049 1,1,4,28,269,3201,44737,711691,12630023,246594988,5244025502,
%T A361049 120540052304,2976918491501,78601791684495,2209667973082374,
%U A361049 65901745111752843,2078619947109354811,69141776287740239348,2419303138068147399700,88842295496847889690405
%N A361049 G.f. satisfies: A(x) = (1/x)*Series_Reversion( x/(1 + x*A(x)^2 + x^2*A(x)*A'(x)) ).
%H A361049 Paul D. Hanna, <a href="/A361049/b361049.txt">Table of n, a(n) for n = 0..300</a>
%F A361049 Given g.f. A(x) = Sum_{n>=0} a(n)*x^n, let G(x) be the g.f. of A361048, then the following formulas hold.
%F A361049 (1) A(x) = (1/x)*Series_Reversion( x/(1 + x*A(x)^2 + x^2*A(x)*A'(x)) ).
%F A361049 (2) A(x) = G(x*A(x)).
%F A361049 (4) A(x) = (1/x)*Series_Reversion(x/G(x)).
%F A361049 (3) G(x) = A(x/G(x)).
%F A361049 (5) G(x) = 1 + d/dx (x^2 * A(x)^2)/2.
%F A361049 a(n) ~ c * n! * n^(3*LambertW(1) - 2 + 3/(1 + LambertW(1))) / LambertW(1)^n, where c = 0.13835030685615842626... - _Vaclav Kotesovec_, Mar 13 2023
%e A361049 G.f.: A(x) = 1 + x + 4*x^2 + 28*x^3 + 269*x^4 + 3201*x^5 + 44737*x^6 + 711691*x^7 + 12630023*x^8 + 246594988*x^9 + ...
%e A361049 such that A(x) = G(x*A(x)) where G(x) is given by
%e A361049 G(x) = 1 + d/dx (x^2 * A(x)^2)/2, which begins
%e A361049 G(x) = 1 + x + 3*x^2 + 18*x^3 + 160*x^4 + 1830*x^5 + 25074*x^6 + 395248*x^7 + 6990876*x^8 + ... + A361048(n)*x^n + ...
%o A361049 (PARI) {a(n) = my(A=1); for(i=1,n,
%o A361049 A = (1/x)*serreverse( x/(1 + x*A^2 + x^2*A*A') +x^2*O(x^n) )); polcoeff(A,n)}
%o A361049 for(n=0,30, print1(a(n),", "))
%Y A361049 Cf. A361048, A132070, A113882.
%K A361049 nonn
%O A361049 0,3
%A A361049 _Paul D. Hanna_, Mar 13 2023