| E.g.f: A(x) = 1 + 3x + 35x^2/2! + 1296x^3/3! + 157437x^4/4! +...
A(x) = W(x) + 2*W(2x)^2*x + 2^4*W(4x)^3*x^2/2! + 2^9*W(8*x)^4*x^3/3! +...
W(x) = LambertW(-x)/(-x) = 1 + x + 3*x^2/2! + 4^2*x^3/3! +...+ (n+1)^(n-1)*x^n/n! +...
This is a special application of the following identity.
Let F(x) be a formal power series in x such that F(0)=1, then
Sum_{n>=0} m^n * F(q^n*x)^b * log( F(q^n*x) )^n / n! =
Sum_{n>=0} x^n * [y^n] F(y)^(m*q^n + b) .
The e.g.f. of this sequence is derived as follows.
Let F(x) = W(x) = LambertW(-x)/(-x);
since log( W(x) ) = x*W(x) and
since W(x)^m = Sum_{n>=0} m*(m+n)^(n-1)*x^n/n! then
Sum_{n>=0} m^n * q^(n^2) * W(q^n*x)^(b+n) * x^n/ n! =
Sum_{n>=0} (m*q^n + b) * (m*q^n + b + n)^(n-1) * x^n.
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