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A195254 O.g.f.: Sum_{n>=0} 2*(n+2)^(n-1)*x^n/(1+n*x)^n.

%I A195254 #13 Aug 17 2013 22:15:01
%S A195254 1,2,6,20,76,336,1744,10592,74400,595712,5362432,53626368,589894144,
%T A195254 7078737920,92023609344,1288330563584,19324958519296,309199336439808,
%U A195254 5256388719738880,94614996955824128,1797684942161707008,35953698843236237312,755027675707965177856
%N A195254 O.g.f.: Sum_{n>=0} 2*(n+2)^(n-1)*x^n/(1+n*x)^n.
%C A195254 Compare the g.f. to: W(x)^2 = Sum_{n>=0} 2*(n+2)^(n-1)*x^n/n! where W(x) = LambertW(-x)/(-x).
%C A195254 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.
%H A195254 Vincenzo Librandi, <a href="/A195254/b195254.txt">Table of n, a(n) for n = 0..200</a>
%F A195254 a(n) = (n-1)!*Sum_{k=1..n} 2^k/(k-1)! for n>0, with a(0)=1.
%F A195254 Recurrence: a(n) = (n+1)*a(n-1) - 2*(n-2)*a(n-2). - _Vaclav Kotesovec_, Aug 17 2013
%F A195254 a(n) ~ 2*exp(2) * (n-1)!. - _Vaclav Kotesovec_, Aug 17 2013
%e A195254 O.g.f.: A(x) = 1 + 2*x + 6*x^2 + 20*x^3 + 76*x^4 + 336*x^5 + 1744*x^6 +...
%e A195254 where
%e A195254 A(x) = 1 + 2*x/(1+x) + 2*4*x^2/(1+2*x)^2 + 2*5^2*x^3/(1+3*x)^3 + 2*6^3*x^4/(1+4*x)^4 +...
%t A195254 Flatten[{1,Table[(n-1)!*Sum[2^k/(k-1)!,{k,1,n}],{n,1,20}]}] (* _Vaclav Kotesovec_, Aug 17 2013 *)
%o A195254 (PARI) {a(n)=polcoeff(sum(m=0,n,2*(m+2)^(m-1)*x^m/(1+m*x+x*O(x^n))^m),n)}
%o A195254 (PARI) {a(n)=if(n==0,1,(n-1)!*sum(k=1,n,2^k/(k-1)!))}
%Y A195254 Cf. A000522, A195255, A195256, A195257.
%K A195254 nonn,easy
%O A195254 0,2
%A A195254 _Paul D. Hanna_, Sep 13 2011