A223076 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n * A(2*n*x)^n/n! * exp(-n*x*A(2*n*x)).

1, 1, 3, 25, 433, 14929, 1009039, 134378493, 35413549073, 18529994604561, 19287258947192299, 39990414610486392193, 165330456559779835205073, 1363910437230335758822062353, 22464490025153709857947688719687, 739043653017364758151896078253911765

Offset: 0

Author

Paul D. Hanna, Mar 14 2013

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 + 3*x^2 + 25*x^3 + 433*x^4 + 14929*x^5 + 1009039*x^6 +...
where
A(x) = 1 + x*A(2*x)*exp(-x*A(2*x)) + 2^2*x^2*A(4*x)^2/2!*exp(-2*x*A(4*x)) + 3^3*x^3*A(6*x)^3/3!*exp(-3*x*A(6*x)) + 4^4*x^4*A(8*x)^4/4!*exp(-4*x*A(8*x)) + 5^5*x^5*A(10*x)^5/5!*exp(-5*x*A(10*x)) +...
simplifies to a power series in x with integer coefficients.
		

Crossrefs

Cf. A223075, A218672, 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, 2*k*x)^k/k!*exp(-k*x*subst(A, x, 2*k*x)+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Formula

a(4*n+2) == 3 (mod 4) for n>=0; a(n) == 1 (mod 2) for n>=0.