A185029 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n * A(n^4*x)^n/n! * exp(-n*x*A(n^4*x)).

1, 1, 2, 65, 3524, 1364432, 1445333132, 7913299718555, 162327934705456532, 14083866155101076361024, 5251111824344114834186373747, 7956883819596423111541696080219295, 51760975171209084256721290749117849746987, 1424616119143714906580708999710589586791029920856

Offset: 0

Author

Paul D. Hanna, Dec 24 2012

Keywords

• nonn

Comments

Compare to the LambertW identity:

Sum_{n>=0} n^n * x^n * G(x)^n/n! * exp(-n*x*G(x)) = 1/(1 - x*G(x)).

Examples

			 O.g.f.: A(x) = 1 + x + 2*x^2 + 65*x^3 + 3524*x^4 + 1364432*x^5 +...
where
A(x) = 1 + x*A(x)*exp(-x*A(x)) + 2^2*x^2*A(2^4*x)^2/2!*exp(-2*x*A(2^4*x)) + 3^3*x^3*A(3^4*x)^3/3!*exp(-3*x*A(3^4*x)) + 4^4*x^4*A(4^4*x)^4/4!*exp(-4*x*A(4^4*x)) + 5^5*x^5*A(5^4*x)^5/5!*exp(-5*x*A(5^4*x)) +...
simplifies to a power series in x with integer coefficients.
		

Crossrefs

Cf. A218672, A218681, A219342.