cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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.

Original entry on oeis.org

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

Views

Author

Washington Bomfim, Mar 18 2012

Keywords

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.

Examples

			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.

References

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

Links

Crossrefs

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

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)))

A210113 Floor of the expected value of number of trials until exactly two cells are empty in a random distribution of n balls in n cells.

Original entry on oeis.org

9, 3, 2, 1, 2, 3, 4, 7, 12, 21, 40, 75, 147, 292, 594, 1229, 2582, 5499, 11859, 25868, 57008, 126814, 284523, 643401, 1465511, 3360493, 7753730, 17993787, 41982506, 98445184, 231932762, 548839352, 1304155087
Offset: 3

Views

Author

Washington Bomfim, Mar 18 2012

Keywords

Comments

Also floor of the expected value of number of trials until we have n-2 distinct symbols in a random sequence on n symbols of length n. A055775 corresponds to zero cells empty.

Examples

			For n=3, there are 3^3 = 27 sequences on 3 symbols of length 3. Only 3 sequences has a unique symbol, so a(3) = floor(27/3) = 9.

References

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

Links

Crossrefs

Cf. A055775, A209899, A209900, A210112, A210114, A210115, A210116.

Formula

With m = 2, a(n) = floor(n^n/(binomial(n,m)*_Sum{v=0..n-m-1}((-1)^v*binomial(n-m,v)*(n-m-v)^n)))

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.

Original entry on oeis.org

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

Views

Author

Washington Bomfim, Mar 18 2012

Keywords

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.

Examples

			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.

References

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

Links

Crossrefs

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

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)))

A210116 Floor of the expected value of number of trials until exactly five cells are empty in a random distribution of n balls in n cells.

Original entry on oeis.org

7776, 311, 51, 16, 7, 4, 3, 3, 2, 3, 3, 4, 5, 8, 11, 16, 25, 40, 66, 110, 187, 325, 574, 1032, 1885, 3492, 6557, 12467, 23988, 46667, 91731, 182078, 364734, 736972, 1501318, 3082136, 6374007, 13273719, 27825438, 58697777, 124566798
Offset: 6

Views

Author

Washington Bomfim, Mar 18 2012

Keywords

Comments

Also floor of the expected value of number of trials until we have n-5 distinct symbols in a random sequence on n symbols of length n. A055775 corresponds to zero cells empty.

Examples

			For n=6, there are 6^6 = 46656 sequences on 6 symbols of length 6. Only 6 sequences has a unique symbol, so a(6) = floor(46656/6) = 7776.

References

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

Links

Crossrefs

Cf. A055775, A209899, A209900, A210112, A210113, A210114, A210115.

Formula

With m = 5, a(n) = floor(n^n/(binomial(n,m)*_Sum{v=0..n-m-1}((-1)^v*binomial(n-m,v)*(n-m-v)^n)))
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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