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%I #7 Oct 10 2020 08:25:30
%S 1,5,30,185,1180,7845,54850,407225,3241200,27882725,260710150,
%T 2655929625,29459366500,354733101125,4617633830250,64677391201625,
%U 970313455915000,15525778234093125,263942044676848750,4750975877669605625,90268637043154147500
%N O.g.f.: Sum_{n>=0} 5*(n+5)^(n-1)*x^n/(1+n*x)^n.
%C Compare the g.f. to: W(x)^5 = Sum_{n>=0} 5*(n+5)^(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} 5^k/(k-1)! for n>0, with a(0)=1.
%F a(n) ~ 5*exp(5) * (n-1)!. - _Vaclav Kotesovec_, Oct 10 2020
%e O.g.f.: A(x) = 1 + 5*x + 30*x^2 + 185*x^3 + 1180*x^4 + 7845*x^5 +...
%e where
%e A(x) = 1 + 5*x/(1+x) + 5*7*x^2/(1+2*x)^2 + 5*8^2*x^3/(1+3*x)^3 + 5*9^3*x^4/(1+4*x)^4 +...
%o (PARI) {a(n)=polcoeff(sum(m=0,n,5*(m+5)^(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,5^k/(k-1)!))}
%Y Cf. A000522, A195254, A195255, A195256.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Sep 13 2011
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