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FORMULA
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G.f. A(x) = Sum_{n>=1} a(n)*x^n may be defined by the following.
(1) A( x - x^3 * A'(x)/A(x) ) = x.
(2) A(x) = x + A(x)^3 * A'(A(x)) / A(A(x)).
(3) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^(3*n) * (A'(x)/A(x))^n / n!.
(4) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(3*n-1) * (A'(x)/A(x))^n / n! ).
(5) B(x) = 1 + x*Sum_{n>=1} d^n/dx^n x^(2*n-1) * B(x)^n / n!, where B(x) = x*A'(x)/A(x) is the g.f. of A356848.
a(n) ~ c * n! / (n^(2*(1 - LambertW(1))) * LambertW(1)^n), where c = 0.23898347792869028031... - Vaclav Kotesovec, Feb 23 2023
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EXAMPLE
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G.f.: A(x) = x + x^2 + 3*x^3 + 15*x^4 + 105*x^5 + 941*x^6 + 10227*x^7 + 130103*x^8 + 1890785*x^9 + 30848357*x^10 + ...
Related series.
Let B(x) = x*A'(x)/A(x), then B(x) is the g.f. of A356848,
B(x) = 1 + x + 5*x^2 + 37*x^3 + 353*x^4 + 4061*x^5 + 54221*x^6 + 820205*x^7 + 13829377*x^8 + 256853629*x^9 + ... + A356848(n)*x^n + ...
such that A( x - x^2*B(x) ) = x,
and B(x) is defined by
B(x) = 1 + x*[(d/dx x*B(x)) + (d^2/dx^2 x^3*B(x)^2)/2! + (d^3/dx^3 x^5*B(x)^3)/3! + (d^4/dx^4 x^7*B(x)^4)/4! + (d^5/dx^5 x^9*B(x)^5)/5! + (d^6/dx^6 x^11*B(x)^6)/6! + ... + (d^n/dx^n x^(2*n-1)*B(x)^n)/n! + ...].
Further,
Series_Reversion(A(x)) = x - x^2 - x^3 - 5*x^4 - 37*x^5 - 353*x^6 - 4061*x^7 - 54221*x^8 - 820205*x^9 + ... + -A356848(n)*x^(n+2) + ...
A(x)^3 = x^3 + 3*x^4 + 12*x^5 + 64*x^6 + 441*x^7 + 3795*x^8 + 39504*x^9 + 483852*x^10 + ...
A'(A(x)) = 1 + 2*x + 11*x^2 + 84*x^3 + 798*x^4 + 9000*x^5 + 117232*x^6 + 1730560*x^7 + 28543340*x^8 + ...
A(A(x)) = x + 2*x^2 + 8*x^3 + 46*x^4 + 342*x^5 + 3118*x^6 + 33730*x^7 + 423014*x^8 + 6042106*x^9 + ...
A'(A(x))/A(A(x)) = 1/x + 3*x + 32*x^2 + 368*x^3 + 4752*x^4 + 68556*x^5 + 1095192*x^6 + 19216988*x^7 + ...
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PROG
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(PARI) {a(n) = my(A=x); for(i=1, n, A=serreverse(x - x^3*A'/A +x*O(x^n))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
(PARI) {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=x); for(i=1, n, A = x + sum(m=1, n, Dx(m-1, x^(3*m)*(A')^m/A^m/m!)) +O(x^(n+1))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
(PARI) {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); for(i=1, n, A = x*exp(sum(m=1, n, Dx(m-1, x^(3*m-1)*(A')^m/A^m/m!)) +O(x^(n+1)))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
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