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A178923
Rectangular array T(m,k)= StirlingS2(k-1,m-1)*m! (The Coupon Collectors Problem)
2
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
OFFSET
1,5
COMMENTS
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.
FORMULA
O.g.f. for row m: m!*x^m/Product_{i=1...m-1}1-i*x.
EXAMPLE
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 ...
MAPLE
A178923 := proc(m, k)
combinat[stirling2](k-1, m-1)*m! ;
end proc:
seq(seq(A178923(m, d-m), m=1..d-1), d=2..15) ; # R. J. Mathar, Jan 19 2024
MATHEMATICA
Table[Table[StirlingS2[k - 1, m - 1] m!, {k, 1, 10}], {m, 1, 10}] // Grid
CROSSREFS
Cf. A068293 (row m=3), A000142 (diagonal), A001804 (subdiagonal).
Sequence in context: A066448 A108497 A108498 * A242609 A226225 A329265
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Dec 29 2010
STATUS
approved