A195242 Expansion of Sum_{n>=0} n^n*x^n/(1 - n*x)^n.

1, 1, 5, 44, 548, 8808, 173352, 4036288, 108507968, 3307368320, 112703108480, 4245680193024, 175200825481728, 7859411394860032, 380810598813553664, 19819617775693512704, 1102737068471914938368, 65316500202537025634304, 4103422475123595857854464

Offset: 0

Author

Paul D. Hanna, Sep 13 2011

Keywords

• nonn

Comments

Compare g.f. to the identity (cf. A001710):

Sum_{n>=0} n^n*x^n/(1 + n*x)^n = 1 + (1/2)*Sum_{n>=1} (n+1)!*x^n.

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 44*x^3 + 548*x^4 + 8808*x^5 + 173352*x^6 +...
where
A(x) = 1 + x/(1-x) + 2^2*x^2/(1-2*x)^2 + 3^3*x^3/(1-3*x)^3 + 4^4*x^4/(1-4*x)^4 +...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..371

Crossrefs

Cf. A001710, A053506, A242449.

Programs

Mathematica

a[n_] := Sum[Binomial[n - 1, k] (k + 1)^n, {k, 0, n}];
Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Jun 26 2019 *)

PARI

{a(n)=polcoeff(sum(m=0,n,m^m*x^m/(1-m*x+x*O(x^n))^m),n)}

PARI

{a(n)=sum(k=0,n,binomial(n-1,k)*(k+1)^n)}

PARI

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

Formula

a(n) = Sum_{k=0..n} C(n-1,k)*(k+1)^n.

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

a(n) ~ n^n * r^(n+3/2) / (exp(n) * (1-r)^n), where r = 1/(1+LambertW(exp(-1))) = 0.78218829428019990122... . - Vaclav Kotesovec, May 14 2014

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(-k,-n)*k^n. Cf. A053506. - Peter Luschny, Apr 11 2016