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

1, 2, 6, 20, 76, 336, 1744, 10592, 74400, 595712, 5362432, 53626368, 589894144, 7078737920, 92023609344, 1288330563584, 19324958519296, 309199336439808, 5256388719738880, 94614996955824128, 1797684942161707008, 35953698843236237312, 755027675707965177856

Offset: 0

Author

Paul D. Hanna, Sep 13 2011

Keywords

• nonn
• easy

Comments

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).

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.

Examples

			O.g.f.: A(x) = 1 + 2*x + 6*x^2 + 20*x^3 + 76*x^4 + 336*x^5 + 1744*x^6 +...
where
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 +...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A000522, A195255, A195256, A195257.

Programs

Mathematica

Flatten[{1,Table[(n-1)!*Sum[2^k/(k-1)!,{k,1,n}],{n,1,20}]}] (* Vaclav Kotesovec, Aug 17 2013 *)

PARI

{a(n)=polcoeff(sum(m=0,n,2*(m+2)^(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,2^k/(k-1)!))}

Formula

a(n) = (n-1)!*Sum_{k=1..n} 2^k/(k-1)! for n>0, with a(0)=1.

Recurrence: a(n) = (n+1)*a(n-1) - 2*(n-2)*a(n-2). - Vaclav Kotesovec, Aug 17 2013

a(n) ~ 2*exp(2) * (n-1)!. - Vaclav Kotesovec, Aug 17 2013