This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A178923 #25 Jun 02 2025 02:58:47
%S A178923 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,
%T A178923 144,0,0,0,0,0,2,186,600,120,0,0,0,0,0,2,378,2160,1200,0,0,0,0,0
%N A178923 Rectangular array T(m,k)= StirlingS2(k-1,m-1)*m! (The Coupon Collectors Problem).
%C A178923 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.
%C A178923 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.
%F A178923 O.g.f. for row m: m!*x^m/Product_{i=1...m-1}1-i*x.
%e A178923 1 0 0 0 0 0 0 0 0 ...
%e A178923 0 2 2 2 2 2 2 2 2 ...
%e A178923 0 0 6 18 42 90 186 378 762 ...
%e A178923 0 0 0 24 144 600 2160 7224 23184 ...
%e A178923 0 0 0 0 120 1200 7800 42000 204120 ...
%e A178923 0 0 0 0 0 720 10800 100800 756000 ...
%e A178923 0 0 0 0 0 0 5040 105840 1340640 ...
%e A178923 0 0 0 0 0 0 0 40320 1128960 ...
%e A178923 0 0 0 0 0 0 0 0 362880 ...
%p A178923 A178923 := proc(m,k)
%p A178923 combinat[stirling2](k-1,m-1)*m! ;
%p A178923 end proc:
%p A178923 seq(seq(A178923(m,d-m),m=1..d-1),d=2..15) ; # _R. J. Mathar_, Jan 19 2024
%t A178923 Table[Table[StirlingS2[k - 1, m - 1] m!, {k, 1, 10}], {m, 1, 10}] // Grid
%Y A178923 Cf. A068293 (row m=3), A000142 (diagonal), A001804 (subdiagonal).
%K A178923 nonn,tabl
%O A178923 1,5
%A A178923 _Geoffrey Critzer_, Dec 29 2010