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

Author

Geoffrey Critzer, Dec 29 2010

Keywords

• nonn
• tabl

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.

Examples

			   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 ...
		

Crossrefs

Cf. A068293 (row m=3), A000142 (diagonal), A001804 (subdiagonal).

Programs

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

Formula

O.g.f. for row m: m!*x^m/Product_{i=1...m-1}1-i*x.