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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).

%I A257894 #13 Nov 12 2023 04:02:37
%S A257894 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 A257894 3661,63,1,1,363,13489,874853,76111,22631,127,1,1,761,726301,336581,
%U A257894 58067611,952525,137845,255,1,1,7129,3144919,129973303,68165041
%N 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).
%H A257894 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 A257894 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 A257894 T(n,k) = Sum_{j=1..n} (-1)^(j-1)*j^(1-k)*C(n,j).
%e A257894 Array of fractions begins:
%e A257894 1,      1,          1,             1,                 1,                    1, ...
%e A257894 1,    3/2,        7/4,          15/8,             31/16,                63/32, ...
%e A257894 1,   11/6,      85/36,       575/216,         3661/1296,           22631/7776, ...
%e A257894 1,  25/12,    415/144,     5845/1728,       76111/20736,        952525/248832, ...
%e A257894 1, 137/60, 12019/3600, 874853/216000, 58067611/12960000, 3673451957/777600000, ...
%e A257894 1,  49/20, 13489/3600,  336581/72000, 68165041/12960000,   483900263/86400000, ...
%e A257894 ...
%e A257894 Row 2 (numerators) is A000225 (Mersenne numbers 2^k-1),
%e A257894 Row 3 is A001240 (Differences of reciprocals of unity),
%e A257894 Row 4 is A028037,
%e A257894 Row 5 is A103878,
%e A257894 Row 6 is not in the OEIS.
%e A257894 Column 2 (numerators) is A001008 (Wolstenholme numbers: numerator of harmonic number),
%e A257894 Column 3 is A027459,
%e A257894 Column 4 is A027462,
%e A257894 Column 5 is A072913,
%e A257894 Column 6 is not in the OEIS.
%t A257894 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 A257894 Cf. A257895 (denominators).
%K A257894 nonn,frac,tabl
%O A257894 1,5
%A A257894 _Jean-François Alcover_, May 12 2015