OFFSET
0,4
COMMENTS
Compare to the LambertW identity:
Sum_{n>=0} (n-1)^n * x^n * G(x)^n/n! * exp(-(n-1)*x*G(x)) = 1/(1 - x*G(x)).
EXAMPLE
O.g.f.: A(x) = 1 + x + x^2 + 3*x^3 + 16*x^4 + 117*x^5 + 1185*x^6 +...
where
A(x) = exp(x) + 0*x*A(x)*exp(-0*x*A(x)) + 1^2*x^2*A(2*x)^2/2!*exp(-1*x*A(2*x)) + 2^3*x^3*A(3*x)^3/3!*exp(-2*x*A(3*x)) + 3^4*x^4*A(4*x)^4/4!*exp(-3*x*A(4*x)) + 4^5*x^5*A(5*x)^5/5!*exp(-4*x*A(5*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-1)^k*x^k*subst(A, x, k*x)^k/k!*exp(-(k-1)*x*subst(A, x, k*x)+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 15 2013
STATUS
approved