OFFSET
0,2
COMMENTS
Compare the g.f. to the LambertW identity:
1 = Sum_{n>=0} 3*(n+3)^(n-1) * exp(-(n+3)*x) * x^n/n!.
EXAMPLE
O.g.f.: A(x) = 1 + 3*x + 27*x^2 + 579*x^3 + 20175*x^4 + 1040841*x^5 +...
where
A(x) = 1 + 3*4^0*1^1*x*A(x)*exp(-4*1*x*A(x)) + 3*5^1*2^2*x^2*A(2*x)^2*exp(-5*2*x*A(2*x))/2! + 3*6^2*3^3*x^3*A(3*x)^3*exp(-6*3*x*A(3*x))/3! + 3*7^3*4^4*x^4*A(4*x)^4*exp(-7*4*x*A(4*x))/4! + 3*8^4*5^5*x^5*A(5*x)^5*exp(-8*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, 3*(k+3)^(k-1)*k^k*x^k*subst(A, x, k*x)^k/k!*exp(-(k+3)*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 18 2012
STATUS
approved