A210112 Floor of the expected value of number of trials until exactly one cell is empty in a random distribution of n balls in n cells.

2, 1, 1, 2, 4, 7, 14, 29, 61, 129, 282, 623, 1400, 3189, 7347, 17101, 40167, 95110, 226841, 544555, 1314983, 3192458, 7788521, 19086807, 46968280, 116019696, 287602234, 715281652, 1784383956, 4464139806

Offset

2,1

Comments

Also floor of the expected value of number of trials until we have n-1 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 = 2..100

Formula

With m = 1, 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=2, with symbols 0 and 1, the 2^2 sequences on 2 symbols of length 2 can be represented by 00, 01, 10, and 11. We have 2 sequences with a unique symbol, so a(2) = floor(4/2) = 2.

Crossrefs

Cf. A055775, A209899, A209900, A210113, A210114, A210115, A210116.

Sequence in context: A352501 A119558 A333450 * A024735 A024957 A153914

Adjacent sequences: A210109 A210110 A210111 * A210113 A210114 A210115

Keyword

nonn

Author

Washington Bomfim, Mar 18 2012

Status

approved