A072913 -id:A072913 - cp's OEIS frontend

A072914 Denominators of 1/4!(H(n,1)^4+6H(n,1)^2H(n,2)+8H(n,1)H(n,3)+3H(n,2)^2+6*H(n,4)), where H(n,m) = Sum_{i=1..n} 1/i^m are generalized harmonic numbers.

1, 16, 1296, 20736, 12960000, 12960000, 31116960000, 497871360000, 40327580160000, 40327580160000, 590436101122560000, 590436101122560000, 16863445484161436160000, 16863445484161436160000

Offset: 1

Vladeta Jovovic, Aug 10 2002

a(n) = A007480 (n) for n=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 26, 27, 51, 52, 53, 54, 110, 111, 112, 113, 114, 115, 116...... - Benoit Cloitre, Aug 13 2002

Cf. A072913.

x(n)=sum(k=1,n,1/k); y(n)=sum(k=1,n,1/k^2); z(n)=sum(k=1,n,1/k^3); w(n)=sum(k=1,n,1/k^4); a(n)=denominator(1/4!*(x(n)^4+6*x(n)^2*y(n)+8*x(n)*z(n)+3*y(n)^2+6*w(n)))

Denominators of 1/4!*((gamma+Psi(n+1))^4+6*(gamma+Psi(n+1))^2*(1/6*Pi^2-Psi(1, n+1))+8*(gamma+Psi(n+1))*(Zeta(3)+1/2*Psi(2, n+1))+3*(1/6*Pi^2-Psi(1, n+1))^2+6*(1/90*Pi^4-1/6*Psi(3, n+1))).

More terms from Benoit Cloitre, Aug 13 2002

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).

Original entry on oeis.org

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

Jean-François Alcover, May 12 2015

			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.

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.

Cf. A257895 (denominators).

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

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

cp's OEIS Frontend

A072914 Denominators of 1/4!(H(n,1)^4+6H(n,1)^2H(n,2)+8H(n,1)H(n,3)+3H(n,2)^2+6*H(n,4)), where H(n,m) = Sum_{i=1..n} 1/i^m are generalized harmonic numbers.

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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).

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cp's OEIS Frontend

A072914 Denominators of 1/4!*(H(n,1)^4+6*H(n,1)^2*H(n,2)+8*H(n,1)*H(n,3)+3*H(n,2)^2+6*H(n,4)), where H(n,m) = Sum_{i=1..n} 1/i^m are generalized harmonic numbers.

Views

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Comments

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Programs

PARI

Formula

Extensions

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).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A072914 Denominators of 1/4!(H(n,1)^4+6H(n,1)^2H(n,2)+8H(n,1)H(n,3)+3H(n,2)^2+6*H(n,4)), where H(n,m) = Sum_{i=1..n} 1/i^m are generalized harmonic numbers.