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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).
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%I #13 Nov 12 2023 04:02:37

%S 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,

%T 3661,63,1,1,363,13489,874853,76111,22631,127,1,1,761,726301,336581,

%U 58067611,952525,137845,255,1,1,7129,3144919,129973303,68165041

%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 (numerators).

%H Zhi-Dong Bai, Chern-Ching Chao, Hsien-Kuei Hwang and Wen-Qi Liang, <a href="http://projecteuclid.org/download/pdf_1/euclid.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)*C(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 (numerators) is A000225 (Mersenne numbers 2^k-1),

%e Row 3 is A001240 (Differences of reciprocals of unity),

%e Row 4 is A028037,

%e Row 5 is A103878,

%e Row 6 is not in the OEIS.

%e Column 2 (numerators) is A001008 (Wolstenholme numbers: numerator of harmonic number),

%e Column 3 is A027459,

%e Column 4 is A027462,

%e Column 5 is A072913,

%e Column 6 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] // Numerator, {n, 1, 12}, {k, 1, n}] // Flatten

%Y Cf. A257895 (denominators).

%K nonn,frac,tabl

%O 1,5

%A _Jean-François Alcover_, May 12 2015