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

0,3

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)).

Links

Table of n, a(n) for n=0..13.

Example

 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.

Prog

 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, k^k*x^k*subst(A, x, k^4*x)^k/k!*exp(-k*x*subst(A, x, k^4*x)+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 16, print1(a(n), ", "))

Crossrefs

Cf. A218672, A218681, A219342.

Sequence in context: A294273 A199145 A198665 * A228081 A214366 A220596

Adjacent sequences: A185026 A185027 A185028 * A185030 A185031 A185032

Keyword

nonn

Author

Paul D. Hanna, Dec 24 2012

Status

approved