OFFSET
0,2
COMMENTS
Compare the g.f. to: W(x)^4 = Sum_{n>=0} 4*(n+4)^(n-1)*x^n/n! where W(x) = LambertW(-x)/(-x).
Compare to a g.f. of A000522: Sum_{n>=0} (n+1)^(n-1)*x^n/(1+n*x)^n, which generates the total number of arrangements of a set with n elements.
FORMULA
a(n) = (n-1)!*Sum_{k=1..n} 4^k/(k-1)! for n>0, with a(0)=1.
a(n) ~ 4*exp(4) * (n-1)!. - Vaclav Kotesovec, Oct 10 2020
EXAMPLE
O.g.f.: A(x) = 1 + 4*x + 20*x^2 + 104*x^3 + 568*x^4 + 3296*x^5 +...
where
A(x) = 1 + 4*x/(1+x) + 4*6*x^2/(1+2*x)^2 + 4*7^2*x^3/(1+3*x)^3 + 4*8^3*x^4/(1+4*x)^4 +...
PROG
(PARI) {a(n)=polcoeff(sum(m=0, n, 4*(m+4)^(m-1)*x^m/(1+m*x+x*O(x^n))^m), n)}
(PARI) {a(n)=if(n==0, 1, (n-1)!*sum(k=1, n, 4^k/(k-1)!))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 13 2011
STATUS
approved