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A360978 G.f. satisfies: A(x) = Series_Reversion(x - x^2*A'(x)^3).

%I A360978 #8 Feb 28 2023 04:26:23
%S A360978 1,1,8,119,2476,64370,1974468,69109563,2702001936,116298977966,
%T A360978 5453395749960,276403464191890,15049886389916756,875933263547340216,
%U A360978 54268470230312961400,3566244291096016078419,247800396100716098128236,18155541676448293842945990
%N A360978 G.f. satisfies: A(x) = Series_Reversion(x - x^2*A'(x)^3).
%C A360978 a(n) = A360975(n-1)/n for n >= 1.
%H A360978 Paul D. Hanna, <a href="/A360978/b360978.txt">Table of n, a(n) for n = 1..300</a>
%F A360978 G.f. A(x) = Sum_{n>=1} a(n)*x^n may be defined by the following.
%F A360978 (1) A(x) = Series_Reversion(x - x^2*A'(x)^3).
%F A360978 (2) A(x) = x + A(x)^2 * A’(A(x))^3.
%F A360978 (3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * A'(x)^(3*n) / n! ).
%F A360978 (4) A'(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * A'(x)^(3*n) / n!.
%e A360978 G.f.: A(x) = x + x^2 + 8*x^3 + 119*x^4 + 2476*x^5 + 64370*x^6 + 1974468*x^7 + 69109563*x^8 + 2702001936*x^9 + ...
%e A360978 By definition, A(x - x^2*A'(x)^3) = x, where
%e A360978 A'(x) = 1 + 2*x + 24*x^2 + 476*x^3 + 12380*x^4 + 386220*x^5 + 13821276*x^6 + 552876504*x^7 + ... + A360975(n)*x^n + ...
%e A360978 Also,
%e A360978 A'(x) = 1 + (d/dx x^2*A'(x)^3) + (d^2/dx^2 x^4*A'(x)^6)/2! + (d^3/dx^3 x^6*A'(x)^9)/3! + (d^4/dx^4 x^8*A'(x)^12)/4! + (d^5/dx^5 x^10*A'(x)^15)/5! + ... + (d^n/dx^n x^(2*n)*A'(x)^(3*n))/n! + ...
%e A360978 Further,
%e A360978 A(x) = x * exp( x*A'(x)^3 + (d/dx x^3*A'(x)^6)/2! + (d^2/dx^2 x^5*A'(x)^9)/3! + (d^3/dx^3 x^7*A'(x)^12)/4! + (d^4/dx^4 x^9*A'(x)^15)/5! + ... + (d^(n-1)/dx^(n-1) x^(2*n-1)*A'(x)^(3*n))/n! + ... ).
%o A360978 (PARI) {a(n) = my(A=x+x^2); for(i=1, n, A=serreverse(x - x^2*A'^3 +x*O(x^(n+1)))); polcoeff(A, n)}
%o A360978 for(n=1, 25, print1(a(n), ", "))
%Y A360978 Cf. A360975, A229619, A360976, A360977.
%K A360978 nonn
%O A360978 1,3
%A A360978 _Paul D. Hanna_, Feb 27 2023