A257894 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 (numerators).
1, 1, 1, 1, 3, 1, 1, 11, 7, 1, 1, 25, 85, 15, 1, 1, 137, 415, 575, 31, 1, 1, 49, 12019, 5845, 3661, 63, 1, 1, 363, 13489, 874853, 76111, 22631, 127, 1, 1, 761, 726301, 336581, 58067611, 952525, 137845, 255, 1, 1, 7129, 3144919, 129973303, 68165041
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 (numerators) is A000225 (Mersenne numbers 2^k-1), Row 3 is A001240 (Differences of reciprocals of unity), Row 4 is A028037, Row 5 is A103878, Row 6 is not in the OEIS. Column 2 (numerators) is A001008 (Wolstenholme numbers: numerator of harmonic number), Column 3 is A027459, Column 4 is A027462, Column 5 is A072913, Column 6 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. A257895 (denominators).
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] // Numerator, {n, 1, 12}, {k, 1, n}] // Flatten
Formula
T(n,k) = Sum_{j=1..n} (-1)^(j-1)*j^(1-k)*C(n,j).