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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, 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
1,5
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.
FORMULA
T(n,k) = Sum_{j=1..n} (-1)^(j-1)*j^(1-k)*binomial(n,j).
EXAMPLE
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.
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
CROSSREFS
Cf. A257894 (numerators).
Sequence in context: A318393 A340128 A139622 * A186023 A103880 A135899
KEYWORD
nonn,frac,tabl
AUTHOR
STATUS
approved