A152170 a(n) is the total size of all the image sets of all functions from [n] to [n]. I.e., a(n) is the sum of the cardinalities of every image set of every function whose domain and co-domain is {1,2,...,n}.

0, 1, 6, 57, 700, 10505, 186186, 3805249, 88099320, 2278824849, 65132155990, 2038428376721, 69332064858420, 2546464715771353, 100444826158022178, 4234886922345707265, 190053371487946575856, 9045570064018726951457, 455099825218118626519470

Offset: 0

Author

Geoffrey Critzer, Nov 27 2008

Keywords

• nonn

Comments

a(n)/n^n is the expected value for the cardinality of the image set of a function that takes [n] to [n].

a(n)/n^(n+1) is the probability that any particular element of [n] will be in the range of a function f : [n] to [n].

Examples

			a(2) = 6 because the image sets of the functions from [2] to [2] are {1},{2},{1,2},{1,2}.
		

Programs

Magma

[n*(n^n-(n-1)^n): n in [0..20]]; // Vincenzo Librandi, Jul 23 2017

Mathematica

Table[Sum[StirlingS2[n, i] i! Binomial[n, i] i, {i, 1, n}], {n, 0, 20}] (* Geoffrey Critzer, Mar 17 2009 *)

Formula

a(n) = n*(n^n - (n-1)^n).

a(n) = Sum_{i=1..n} S(n,i)*i!*binomial(n,i)*i where S(n,i) is the Stirling number of the second kind.

a(n) = Sum_{k=1..n} A090657(n,k)*k.

Limit_{n->infinity} a(n)/n^(n+1) = (e-1)/e. - Thomas Dybdahl Ahle, Apr 24 2011

Extensions

More terms from Geoffrey Critzer, Mar 17 2009