A210115 Floor of the expected value of number of trials until exactly four cells are empty in a random distribution of n balls in n cells.

625, 50, 13, 5, 3, 2, 2, 2, 3, 4, 5, 7, 11, 17, 28, 46, 78, 136, 242, 441, 815, 1533, 2927, 5669, 11123, 22090, 44363, 90027, 184482, 381499, 795686, 1672914, 3543925, 7561129, 16240832, 35106812, 76346759, 166982782, 367206632, 811693449

Offset

5,1

Comments

Also floor of the expected value of number of trials until we have n-4 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

W. Bomfim, Table of n, a(n) for n = 5..100

Formula

With m = 4, 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=5, there are 5^5 = 3125 sequences on 5 symbols of length 5. Only 5 sequences has a unique symbol, so a(5) = floor(3125/5) = 625.

Crossrefs

Cf. A055775, A209899, A209900, A210112, A210113, A210114, A210116.

Sequence in context: A057012 A115484 A005081 * A186484 A184037 A186483

Adjacent sequences: A210112 A210113 A210114 * A210116 A210117 A210118

Keyword

nonn

Author

Washington Bomfim, Mar 18 2012

Status

approved