A086331 Expansion of e.g.f. exp(x)/(1 + LambertW(-x)).

1, 2, 7, 43, 393, 4721, 69853, 1225757, 24866481, 572410513, 14738647221, 419682895325, 13094075689225, 444198818128313, 16278315877572141, 640854237634448101, 26973655480577228769, 1208724395795734172705, 57453178877303382607717, 2887169565412587866031533

Offset: 0

Author

Vladeta Jovovic, Sep 01 2003

Keywords

• nonn

Comments

Binomial transform of A000312. - Tilman Neumann, Dec 13 2008

a(n) is the number of partial functions on {1,2,...,n} that are endofunctions. See comments in A000169 and A126285 by Franklin T. Adams-Watters. - Geoffrey Critzer, Dec 19 2011

Examples

			a(2) = 7 because {}->{}, 1->1, 2->2, and the four functions from {1,2} into {1,2}. Note A000169(2) = 9 because it counts these 7 and 1->2, 2->1.
		

Links

• Winston de Greef, Table of n, a(n) for n = 0..385 (first 201 terms from Vincenzo Librandi)
• V. Kotesovec, Interesting asymptotic formulas for binomial sums, Jun 09 2013

Crossrefs

Cf. A069856, A204042, A277454, A277456, A323280.

Programs

Maple

a:= n-> add(binomial(n,k)*k^k, k=0..n):
seq(a(n), n=0..25);  # Alois P. Heinz, Dec 30 2021

Mathematica

nn=10;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Range[0,nn]!CoefficientList[Series[Exp[x]/(1-t),{x,0,nn}],x]  (* Geoffrey Critzer, Dec 19 2011 *)

PARI

a(n) = sum(k=0,n, binomial(n, k)*k^k ); \\ Joerg Arndt, May 10 2013

PARI

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x)^k/(1-x)^(k+1))) \\ Seiichi Manyama, Jul 04 2022

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x)*sum(k=0, N, (k*x)^k/k!))) \\ Seiichi Manyama, Jul 04 2022

Formula

a(n) = Sum_{k=0..n} binomial(n,k)*k^k.

a(n) ~ e^(1/e)*n^n * (1 + 1/(2*e*n)) ~ 1.444667861... * n^n. - Vaclav Kotesovec, Nov 27 2012

G.f.: Sum_{k>=0} (k * x)^k/(1 - x)^(k+1). - Seiichi Manyama, Jul 04 2022