A210114 Floor of the expected value of number of trials until exactly three cells are empty in a random distribution of n balls in n cells.

64, 10, 4, 2, 2, 2, 2, 3, 5, 7, 11, 18, 31, 55, 100, 185, 348, 670, 1311, 2606, 5254, 10734, 22196, 46407, 98023, 209009, 449580, 974963, 2130442, 4688533, 10387113, 23156162, 51926745, 117090391, 265413053

Offset

4,1

Comments

Also floor of the expected value of number of trials until we have n-3 distinct symbols in a random sequence on n symbols of length n. A055775 corresponds to zero cells empty.

References

W. Feller, An Introduction to Probability Theory and its Applications, 2nd ed, Wiley, New York, 1965, (2.4) p. 92. (Occupancy problems)

Links

Washington Bomfim, Table of n, a(n) for n = 4..100

Formula

With m = 3, a(n) = floor(n^n/(binomial(n,m)*_Sum{v=0..n-m-1}((-1)^v*binomial(n-m,v)*(n-m-v)^n)))

Example

For n=4, there are 4^4 = 256 sequences on 4 symbols of length 4. Only 4 sequences have a unique symbol, so a(4) = floor(256/4) = 64.

Crossrefs

Cf. A055775, A209899, A209900, A210112, A210113, A210115, A210116.

Sequence in context: A288923 A123964 A298923 * A236179 A226044 A065790

Adjacent sequences: A210111 A210112 A210113 * A210115 A210116 A210117

Keyword

nonn

Author

Washington Bomfim, Mar 18 2012

Status

approved