A209081 Floor(A152170(n)/n^n). Floor of the expected value of the cardinality of the image of a function from [n] to [n].

1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 9, 9, 10, 10, 11, 12, 12, 13, 14, 14, 15, 15, 16, 17, 17, 18, 19, 19, 20, 21, 21, 22, 22, 23, 24, 24, 25, 26, 26, 27, 27, 28, 29, 29, 30, 31, 31, 32, 33, 33, 34, 34, 35, 36, 36, 37, 38

Offset: 1

Author

Washington Bomfim, Mar 05 2012

Keywords

• nonn

Comments

From the first commentary of A152170, a(n)= floor(A152170(n)/n^n) = floor((n(n^n-(n-1)^n))/n^n) = floor(n-(n-1)^n/n^(n-1)).

Examples

			a(1) = 1 because the image of a function from [1] to [1] has one value. a(2) = 1 since we can consider functions with domain {x,y}, and image {X,Y}. We can have f(x)=X, f(y)=X; f(x)=X, f(y)=Y; f(x)=Y, f(y)=Y; f(x)=Y, f(y)=X.
The sum of the cardinalities of the images divided by the number of functions is (1+2+1+2)/4 = 1.5. Floor(1.5)=1.
		

Links

• Washington Bomfim, A method to find bijections from a set of n integers to {0,1, ... ,n-1}

Crossrefs

Cf. A152170.

Formula

a(n) = floor(n-(n-1)^n/n^(n-1)).