A195256 O.g.f.: Sum_{n>=0} 4*(n+4)^(n-1)*x^n/(1+n*x)^n.

1, 4, 20, 104, 568, 3296, 20576, 139840, 1044416, 8617472, 78605824, 790252544, 8709555200, 104581771264, 1359831461888, 19038714208256, 285585008091136, 4569377309327360, 77679482978041856, 1398230968482660352, 26566389500682174464

Offset: 0

Author

Paul D. Hanna, Sep 13 2011

Keywords

• nonn

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.

Examples

			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 +...
		

Crossrefs

Cf. A000522, A195254, A195255, A195257.

Programs

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)!))}

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