A178923 Rectangular array T(m,k)= StirlingS2(k-1,m-1)*m! (The Coupon Collectors Problem)

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.

Links

Table of n, a(n) for n=1..55.

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

Adjacent sequences: A178920 A178921 A178922 * A178924 A178925 A178926

Keyword

nonn,tabl

Author

Geoffrey Critzer, Dec 29 2010

Status

approved