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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

%I

%S 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,

%T 1296,32,1,1,140,3600,216000,20736,7776,64,1,1,280,176400,72000,

%U 12960000,248832,46656,128,1,1,2520,705600,24696000,12960000,777600000

%N 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).

%H Zhi-Dong Bai, Chern-Ching Chao, Hsien-Kuei Hwang and Wen-Qi Liang, <a href="https://doi.org/10.1214/aoap/1028903455">On the variance of the number of maxima in random vectors and its applications</a>, The Annals of Applied Probability 1998, Vol. 8, No. 3, 886-895.

%H O. E. Barndorff-Nielsen and M. Sobel, <a href="http://www.mathnet.ru/links/2d44785a77c46910741a6ce707ad4c3b/tvp624.pdf">On the distribution of the number of admissible points in a vector random sample.</a> Theory Probab. Appl. 11 249-269.

%F T(n,k) = Sum_{j=1..n} (-1)^(j-1)*j^(1-k)*binomial(n,j).

%e Array of fractions begins:

%e 1, 1, 1, 1, 1, 1, ...

%e 1, 3/2, 7/4, 15/8, 31/16, 63/32, ...

%e 1, 11/6, 85/36, 575/216, 3661/1296, 22631/7776, ...

%e 1, 25/12, 415/144, 5845/1728, 76111/20736, 952525/248832, ...

%e 1, 137/60, 12019/3600, 874853/216000, 58067611/12960000, 3673451957/777600000, ...

%e 1, 49/20, 13489/3600, 336581/72000, 68165041/12960000, 483900263/86400000, ...

%e ...

%e Row 2 (denominators) is A000079 (powers of 2),

%e Row 3 is A000400 (powers of 6),

%e Row 4 is A001021 (powers of 12),

%e Row 5 is A159991,

%e Row 6 is not in the OEIS.

%e Column 2 (denominators) is A002805 (denominators of harmonic numbers),

%e Column 3 is A051418 (lcm(1..n)^2),

%e Column 4 is not in the OEIS.

%t 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

%Y Cf. A257894 (numerators).

%K nonn,frac,tabl

%O 1,5

%A _Jean-Fran├žois Alcover_, May 12 2015

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Last modified September 20 16:05 EDT 2020. Contains 337265 sequences. (Running on oeis4.)