A257895 Square array read by ascending antidiagonals where T(n,k) is the mean number of maxima in a set of n random k-dimensional real vectors (denominators).
1, 1, 1, 1, 2, 1, 1, 6, 4, 1, 1, 12, 36, 8, 1, 1, 60, 144, 216, 16, 1, 1, 20, 3600, 1728, 1296, 32, 1, 1, 140, 3600, 216000, 20736, 7776, 64, 1, 1, 280, 176400, 72000, 12960000, 248832, 46656, 128, 1, 1, 2520, 705600, 24696000, 12960000, 777600000
Offset: 1
Examples
Array of fractions begins: 1, 1, 1, 1, 1, 1, ... 1, 3/2, 7/4, 15/8, 31/16, 63/32, ... 1, 11/6, 85/36, 575/216, 3661/1296, 22631/7776, ... 1, 25/12, 415/144, 5845/1728, 76111/20736, 952525/248832, ... 1, 137/60, 12019/3600, 874853/216000, 58067611/12960000, 3673451957/777600000, ... 1, 49/20, 13489/3600, 336581/72000, 68165041/12960000, 483900263/86400000, ... ... Row 2 (denominators) is A000079 (powers of 2), Row 3 is A000400 (powers of 6), Row 4 is A001021 (powers of 12), Row 5 is A159991, Row 6 is not in the OEIS. Column 2 (denominators) is A002805 (denominators of harmonic numbers), Column 3 is A051418 (lcm(1..n)^2), Column 4 is not in the OEIS.
Links
- Zhi-Dong Bai, Chern-Ching Chao, Hsien-Kuei Hwang and Wen-Qi Liang, On the variance of the number of maxima in random vectors and its applications, The Annals of Applied Probability 1998, Vol. 8, No. 3, 886-895.
- O. E. Barndorff-Nielsen and M. Sobel, On the distribution of the number of admissible points in a vector random sample. Theory Probab. Appl. 11 249-269.
Crossrefs
Cf. A257894 (numerators).
Programs
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Mathematica
T[n_, k_] := Sum[(-1)^(j - 1)*j^(1 - k)*Binomial[n, j], {j, 1, n}]; Table[T[n - k + 1, k] // Denominator, {n, 1, 12}, {k, 1, n}] // Flatten
Formula
T(n,k) = Sum_{j=1..n} (-1)^(j-1)*j^(1-k)*binomial(n,j).