A209276 O.g.f. satisfies: A(x) = Sum_{n>=0} (n+2)^n * x^n * A((n+2)*x)^n/n! * exp(-(n+2)*x*A((n+2)*x)).

1, 1, 6, 133, 9403, 2065969, 1400088539, 2908156231705, 18410003437367130, 353588715425938097698, 20534146782689861283550052, 3596867485365965032072729708845, 1897112888731795684931545113460297299, 3009299517165127420220975531888408947667944

Offset: 0

Author

Paul D. Hanna, Jan 15 2013

Keywords

• nonn

Comments

Compare to the LambertW identity:

Sum_{n>=0} (n+2)^n * x^n * G(x)^n/n! * exp(-(n+2)*x*G(x)) = 1/(1 - x*G(x)).

Examples

			O.g.f.: A(x) = 1 + x + 6*x^2 + 133*x^3 + 9403*x^4 + 2065969*x^5 +...
where
A(x) = exp(-2*x*A(2*x)) + 3*x*A(3*x)*exp(-3*x*A(3*x)) + 4^2*x^2*A(4*x)^2/2!*exp(-4*x*A(4*x)) + 5^3*x^3*A(5*x)^3/3!*exp(-5*x*A(5*x)) + 6^4*x^4*A(6*x)^4/4!*exp(-6*x*A(6*x)) + 7^5*x^5*A(7*x)^5/5!*exp(-7*x*A(7*x)) +...
simplifies to a power series in x with integer coefficients.
		

Crossrefs

Cf. A218672, A193363, A209277.

Programs

PARI

{a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, (k+2)^k*x^k*subst(A, x, (k+2)*x)^k/k!*exp(-(k+2)*x*subst(A, x, (k+2)*x)+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))