OFFSET
1,3
FORMULA
a(n) = n^(n-1) - (n-1)!*[ Sum_{d divides n, d<n} d*( a(d)/d! )^(n/d) ] for n>1 with a(1)=1.
G.f.: Sum_{n>=1} -log(1 - a(n)*x^n/n!) = Sum_{n>=1} n^(n-1)*x^n/n! = -LambertW(-x).
G.f.: Sum_{n>=1} -log(1 - a(n)*exp(-n*x)*x^n/n!) = x.
G.f.: Sum_{n>=1} n*a(n)*x^n/[n! - a(n)*x^n] = Sum_{n>=1} n^n*x^n/n!.
G.f.: Sum_{n>=1} n*a(n)*x^n/[n!*exp(nx) - a(n)*x^n] = x/(1-x).
EXAMPLE
G.f.: W(x) = 1/[(1-x)*(1-x^2/2!)*(1-7*x^3/3!)*(1-55*x^4/4!)*(1-601*x^5/5!)*...]
where W(x) = 1 + x + 3*x^2/2! + 4^2*x^3/3! + 5^3*x^4/4! + 6^4*x^5/5! + ...
and W(x/exp(x)) = exp(x) and exp(x*W(x)) = W(x) = LambertW(-x)/(-x).
MATHEMATICA
a[n_] := a[n] = n^(n-1) - (n-1)! * DivisorSum[n, #*(a[#]/#!)^(n/#) &, #<n &]; Array[a, 20] (* Amiram Eldar, Aug 18 2023 *)
PROG
(PARI) {a(n)=if(n<1, 0, if(n==1, 1, n^(n-1) - (n-1)!*sumdiv(n, d, if(d<n, d*(a(d)/d!)^(n/d)))))}
(PARI) {a(n)=if(n<1, 0, n!*polcoeff(sum(k=0, n+1, (k+1)^(k-1)*x^k/k!)*prod(k=1, n-1, 1-a(k)*x^k/k! +x*O(x^n)), n))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladeta Jovovic and Paul D. Hanna, Apr 17 2009
EXTENSIONS
a(19)-a(20) from Amiram Eldar, Aug 18 2023
STATUS
approved