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A178923
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Rectangular array T(m,k)= StirlingS2(k-1,m-1)*m! (The Coupon Collectors Problem)
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2
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1, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 2, 6, 0, 0, 0, 2, 18, 0, 0, 0, 0, 2, 42, 24, 0, 0, 0, 0, 2, 90, 144, 0, 0, 0, 0, 0, 2, 186, 600, 120, 0, 0, 0, 0, 0, 2, 378, 2160, 1200, 0, 0, 0, 0, 0
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OFFSET
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1,5
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COMMENTS
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T(m,k) is the number of functions f:{1,2,...}->(1,2,...,m} such that the image of f[{1,2,...,k}] is {1,2,...,m} but the image of f[{1,2,...,k-1}] is not.
T(m,k)/m^k is the probability that a collector of m different objects will require exactly k trials (uniform random selection with replacement) to complete the collection.
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LINKS
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FORMULA
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O.g.f.: for row m: m!x^m/Product_{i=1...m-1}1-i*x
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EXAMPLE
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1 0 0 0 0 0 0 0 0 ...
0 2 2 2 2 2 2 2 2 ...
0 0 6 18 42 90 186 378 762 ...
0 0 0 24 144 600 2160 7224 23184 ...
0 0 0 0 120 1200 7800 42000 204120 ...
0 0 0 0 0 720 10800 100800 756000 ...
0 0 0 0 0 0 5040 105840 1340640 ...
0 0 0 0 0 0 0 40320 1128960 ...
0 0 0 0 0 0 0 0 362880 ...
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MAPLE
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combinat[stirling2](k-1, m-1)*m! ;
end proc:
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MATHEMATICA
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Table[Table[StirlingS2[k - 1, m - 1] m!, {k, 1, 10}], {m, 1, 10}] // Grid
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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