OFFSET
1,8
COMMENTS
Problem suggested by Brandon Zeidler. To motivate this sequence, suppose that when objects are placed in the same box, they mix and the information they contain is lost. The sequence tells us how much information we can expect to recover.
FORMULA
a(n, k) = k*(1 - 1/n)^(k - 1). Let f(n, k, i) be the number of assignments such that exactly i boxes have exactly one object. For i > n, f(n, k, i) = 0. For i = k <= n, f(n, k, i) = n!/(n-k)!. Otherwise, f(n, k, i) = sum_{j = 1..min(floor((k-i)/2), n-i) A008299(k-i, j)*n!*binomial(k, i)/(n-i-j)!. Then a(n, k) = sum_{i=1..min(n, k)} i*f(n, k, i)/n^k.
EXAMPLE
Array begins:
1 0 0 0 0 0
1 1 3/4 1/2 5/16 3/16
1 4/3 4/3 32/27 80/81 64/81
CROSSREFS
KEYWORD
AUTHOR
David Wasserman, Jun 15 2007
STATUS
approved