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A360977 G.f. satisfies: A(x) = Series_Reversion(x - x^2*A'(x)^2).

%I A360977 #15 Feb 28 2023 04:26:15
%S A360977 1,1,6,65,978,18354,406396,10274553,290411050,9045087486,307195803380,
%T A360977 11287142857810,445880114920868,18841715641048244,848104833677384856,
%U A360977 40515633401768290017,2047629885363936119690,109168898899192961652766,6124201250273921324876116
%N A360977 G.f. satisfies: A(x) = Series_Reversion(x - x^2*A'(x)^2).
%C A360977 a(n) = A360974(n-1)/n for n >= 1.
%H A360977 Paul D. Hanna, <a href="/A360977/b360977.txt">Table of n, a(n) for n = 1..300</a>
%F A360977 G.f. A(x) = Sum_{n>=1} a(n)*x^n may be defined by the following.
%F A360977 (1) A(x) = Series_Reversion(x - x^2*A'(x)^2).
%F A360977 (2) A(x) = x + A(x)^2 * A'(A(x))^2.
%F A360977 (3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * A'(x)^(2*n) / n! ).
%F A360977 (4) A'(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * A'(x)^(2*n) / n!.
%e A360977 G.f.: A(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 + ...
%e A360977 By definition, A(x - x^2*A'(x)^2) = x, where
%e A360977 A'(x) = 1 + 2*x + 18*x^2 + 260*x^3 + 4890*x^4 + 110124*x^5 + 2844772*x^6 + 82196424*x^7 + ... + A360974(n)*x^n + ...
%e A360977 Also,
%e A360977 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^n/dx^n x^(2*n)*A'(x)^(2*n))/n! + ...
%e A360977 Further,
%e A360977 A(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^(n-1)/dx^(n-1) x^(2*n-1)*A'(x)^(2*n))/n! + ... ).
%o A360977 (PARI) {a(n) = my(A=x+x^2); for(i=1, n, A=serreverse(x - x^2*A'^2 +x*O(x^(n+1)))); polcoeff(A, n)}
%o A360977 for(n=1, 25, print1(a(n), ", "))
%Y A360977 Cf. A360974, A229619, A360976, A360978.
%K A360977 nonn
%O A360977 1,3
%A A360977 _Paul D. Hanna_, Feb 27 2023