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A111097
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Maximum likelihood estimate of the number of distinguishable marbles in an urn if repeated random sampling of one marble with replacement yields n different marbles before the first repeated marble.
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0
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1, 2, 5, 8, 13, 19, 25, 33, 42, 51, 62, 74, 86, 100, 115, 130, 147, 165, 183, 203, 224, 245, 268, 292, 316, 342, 369, 396, 425
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| The numbers in my sequence match the first 8 nonzero numbers in sequence A083704 and the first 40 terms (at least) remain very close to those in A083704 . The program below is very crude and will yield the first 20 terms. To get further terms, one must increase the maximum value of n and then increase the maximum value of k so that the maximum value of k is larger than a(n) for the maximum value of n.
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FORMULA
| No formula yet, but I want to stake my claim.
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EXAMPLE
| a(3)=5 because of all urns containing marbles numbered 1, 2, 3, ..., k, an urn containing k = 5 marbles has the largest probability of yielding 3 different marbles before a first repeated marble when sampling with replacement.
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MATHEMATICA
| thetable = Table[N[n/k*Product[(k - i)/k, {i, 1, n - 1}]], {n, 1, 20}, {k, 1, 300}]; maximums = Map[Max, thetable]; maximumlikelihoodestimates = {}; For[i = 1, i <= Length[thetable], i++, maximumlikelihoodestimates = Append[maximumlikelihoodestimates, Position[thetable[[i]], maximums[[i]]]]]; maximumlikelihoodestimates
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CROSSREFS
| Cf. A027916. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 18 2008]
Sequence in context: A200274 A122221 A083704 * A027916 A054254 A025216
Adjacent sequences: A111094 A111095 A111096 * A111098 A111099 A111100
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KEYWORD
| nonn
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AUTHOR
| Marc A. Brodie (mbrodie(AT)wju.edu), Oct 13 2005
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