OFFSET
0,2
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
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
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
EXAMPLE
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.
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladeta Jovovic, Sep 01 2003
STATUS
approved