A368563 - OEIS

%I #20 Jan 22 2024 06:27:14

%S 1,1,2,1,3,5,1,4,4,8,1,5,6,9,13,1,6,7,10,14,19,1,7,8,7,12,18,25,1,8,9,

%T 11,15,20,26,33,1,9,3,12,11,17,27,34,42,1,10,10,6,16,21,24,32,41,51,1,

%U 11,11,13,17,22,28,35,43,52,62,1,12,12,14,10,16,23

%N Table read by downward antidiagonals: ranked maximum r likelihood of the number of distinguishable marbles in an urn if repeated random sampling of one marble with replacement yields n different marbles before the first repeated marble.

%e Table T(n,r) begins:

%e n\r |  1   2   3   4   5   6   7   8   9  10  11

%e ----+-------------------------------------------

%e   1 |  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1

%e   2 |  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12

%e   3 |  5,  4,  6,  7,  8,  9,  3, 10, 11, 12, 13

%e   4 |  8,  9, 10,  7, 11, 12,  6, 13, 14, 15, 16

%e   5 | 13, 14, 12, 15, 11, 16, 17, 10, 18, 19, 20

%e   6 | 19, 18, 20, 17, 21, 22, 16, 23, 24, 15, 25

%e   7 | 25, 26, 27, 24, 28, 23, 29, 22, 30, 31, 21

%e   8 | 33, 34, 32, 35, 31, 36, 30, 37, 38, 29, 39

%e   9 | 42, 41, 43, 40, 44, 39, 45, 46, 38, 47, 37

%e  10 | 51, 52, 50, 53, 54, 49, 55, 48, 56, 47, 57

%e  11 | 62, 63, 61, 64, 60, 65, 59, 66, 58, 67, 57

%e  12 | 74, 73, 75, 72, 76, 71, 77, 70, 78, 79, 69

%e  13 | 86, 87, 85, 88, 84, 89, 90, 83, 91, 82, 92

%t l[n_, r_] := l[n, r] = Table[Product[(M - i + 1)/M, {i, 2, n}]*n/M, {M, 2, n^2*r}];

%t T[1, r_] = 1; T[n_, r_] := T[n, r] = Module[{pos}, pos = Position[l[n, r], RankedMax[l[n, r], r]]; If[Length[pos] == 0, -1, pos[[1, 1]] + 1]];

%t (*Table*)Table[T[n, r], {n, 1, 13}, {r, 1, 11}] // TableForm

%t (*Data*)Table[T[n, r - n + 1], {r, 1, 12}, {n, 1, r}] // Flatten

%Y Cf. A111097 (column 1).

%Y Cf. A000012 (row 1), A000027 (row 2).

%K nonn,tabl

%O 1,3

%A _Robert P. P. McKone_, Dec 30 2023