OFFSET
0,3
COMMENTS
Compare the g.f. to the LambertW identity:
1 = Sum_{n>=0} (n+1)^(n-1) * exp(-(n+1)*x) * x^n/n!.
EXAMPLE
O.g.f.: A(x) = 1 + x + 5*x^2 + 63*x^3 + 1319*x^4 + 40559*x^5 +...
where
A(x) = 1 + 2^0*1^1*x*A(x)*exp(-2*1*x*A(x)) + 3^1*2^2*x^2*A(2*x)^2*exp(-3*2*x*A(2*x))/2! + 4^2*3^3*x^3*A(3*x)^3*exp(-4*3*x*A(3*x))/3! + 5^3*4^4*x^4*A(4*x)^4*exp(-5*4*x*A(4*x))/4! + 6^4*5^5*x^5*A(5*x)^5*exp(-6*5*x*A(5*x))/5! +...
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-1)*k^k*x^k*subst(A, x, k*x)^k/k!*exp(-(k+1)*k*x*subst(A, x, k*x)+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 16 2012
STATUS
approved