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%I #7 Oct 10 2020 08:25:34
%S 1,4,20,104,568,3296,20576,139840,1044416,8617472,78605824,790252544,
%T 8709555200,104581771264,1359831461888,19038714208256,285585008091136,
%U 4569377309327360,77679482978041856,1398230968482660352,26566389500682174464
%N O.g.f.: Sum_{n>=0} 4*(n+4)^(n-1)*x^n/(1+n*x)^n.
%C 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).
%C 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.
%F a(n) = (n-1)!*Sum_{k=1..n} 4^k/(k-1)! for n>0, with a(0)=1.
%F a(n) ~ 4*exp(4) * (n-1)!. - _Vaclav Kotesovec_, Oct 10 2020
%e O.g.f.: A(x) = 1 + 4*x + 20*x^2 + 104*x^3 + 568*x^4 + 3296*x^5 +...
%e where
%e 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 +...
%o (PARI) {a(n)=polcoeff(sum(m=0,n,4*(m+4)^(m-1)*x^m/(1+m*x+x*O(x^n))^m),n)}
%o (PARI) {a(n)=if(n==0,1,(n-1)!*sum(k=1,n,4^k/(k-1)!))}
%Y Cf. A000522, A195254, A195255, A195257.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Sep 13 2011
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