A219342 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n * A(n^3*x)^n/n! * exp(-n*x*A(n^3*x)).

1, 1, 2, 33, 939, 101175, 26230876, 21032800086, 48319626581926, 319633065306440005, 6299181667747767151873, 359980854813102654362716667, 60552379844778585329083453881153, 30125614945616982039421647789900799744, 43971297878008421196972637327280065832735828

Offset: 0

Author

Paul D. Hanna, Nov 18 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 + 33*x^3 + 939*x^4 + 101175*x^5 + 26230876*x^6 +...
where
A(x) = 1 + x*A(x)*exp(-x*A(x)) + 2^2*x^2*A(2^3*x)^2/2!*exp(-2*x*A(2^3*x)) + 3^3*x^3*A(3^3*x)^3/3!*exp(-3*x*A(3^3*x)) + 4^4*x^4*A(4^3*x)^4/4!*exp(-4*x*A(4^3*x)) + 5^5*x^5*A(5^3*x)^5/5!*exp(-5*x*A(5^3*x)) +...
simplifies to a power series in x with integer coefficients.
		

Crossrefs

Cf. A218672, A218681, A185029, A219343, A217900.

Programs

PARI

{a(n)=local(A=1+x);for(i=1,n,A=sum(k=0,n,k^k*x^k*subst(A,x,k^3*x)^k/k!*exp(-k*x*subst(A,x,k^3*x)+x*O(x^n))));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))