A195255 O.g.f.: Sum_{n>=0} 3*(n+3)^(n-1)*x^n/(1+n*x)^n.

1, 3, 12, 51, 234, 1179, 6624, 41931, 300078, 2420307, 21841812, 218595267, 2405079378, 28862546859, 375217892136, 5253064838811, 78796015628886, 1260736379202339, 21432518833860252, 385785340171746003, 7329921466749958458, 146598429345459522363

Offset: 0

Author

Paul D. Hanna, Sep 13 2011

Keywords

• nonn

Comments

Compare the g.f. to: W(x)^3 = Sum_{n>=0} 3*(n+3)^(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 + 3*x + 12*x^2 + 51*x^3 + 234*x^4 + 1179*x^5 +...
where
A(x) = 1 + 3*x/(1+x) + 3*5*x^2/(1+2*x)^2 + 3*6^2*x^3/(1+3*x)^3 + 3*7^3*x^4/(1+4*x)^4 +..
		

Crossrefs

Cf. A000522, A195254, A195256, A195257.

Programs

PARI

{a(n)=polcoeff(sum(m=0,n,3*(m+3)^(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,3^k/(k-1)!))}

Formula

a(n) = (n-1)!*Sum_{k=1..n} 3^k/(k-1)! for n>0, with a(0)=1.

a(n) ~ 3*exp(3) * (n-1)!. - Vaclav Kotesovec, Oct 10 2020