A027459 -id:A027459 - cp's OEIS frontend

A027462 a(n) is the numerator of (-1/6) * Integral_{x=0..1} x^n * log^3(1-x).

1, 15, 575, 5845, 874853, 336581, 129973303, 1149858589, 101622655189, 21945415349, 31276937512951, 33264031387717, 77287019174361937, 81347802723340093, 17055178843123409, 142531324182321979

Offset: 1

Olivier Gérard

nonn

Originally defined as the first column of A027448, but now contains numerator in reduced form (cf. A329122). - Sean A. Irvine, Nov 05 2019

Donal F. Connon, Some series and integrals involving the Riemann zeta function, binomial coefficients and the harmonic numbers. Vol. II(a), arXiv:0710.4023 [math.HO], 2007-2008.
Jerry Metzger and Thomas Richards, A Prisoner Problem Variation, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.7.

Cf. A027459.

a[n_] := -1/6 Integrate[x^(n-1) Log[1-x]^3, {x, 0, 1}] // Numerator;
Table[a[n], {n, 1, 16}] (* Jean-François Alcover, Aug 06 2018 *)

Numerators of sequence a(1,n) in (a(i,j))^4 where a(i,j) = 1/i if j <= i, 0 if j > i.
Numerators of (H(n,1)^3 + 3*H(n,1)*H(n,2) + 2*H(n,3))/(6*n) = ((gamma+Psi(n+1))^3 + 3*(gamma+Psi(n+1))*(1/6*Pi^2 - Psi(1, n+1)) + 2*Zeta(3) + Psi(2, n+1))/(6*n), where H(n, m) = Sum_{i=1..n} 1/i^m are generalized harmonic numbers. - Vladeta Jovovic, Aug 10 2002
From Groux Roland, Feb 11 2011: (Start)
For n>=1, (H(n,1)^3 + 3*H(n,1)*H(n,2) + 2*H(n,3))/(6*n) = -(1/6)*Integral_{x=0..1} x^(n-1)*log^3(1-x) dx = (1/n)*Sum_{j=1..n} (H(n,1) - H(j-1,1))*H(j,1)/j.
For every k>=1 the first column of (a(i,j))^k is the binomial transform of (-1)^n/(n+1)^k.
Proof: the sequence S(n,k) = ((-1)^k/k!)*Integral_{x=0..1} x^(n-1)*log^k(1-x) dx gives the binomial transform of (-1)^n/(n+1)^(k+1) and can be evaluated by parts with S(n,k) = (1/n)*Sum_{j=1..k} S(j,k-1) according to the generation of the first column of (a(i,j))^k.
(End)

Corrected by Vladeta Jovovic, Aug 10 2002

Title changed by Sean A. Irvine, Nov 05 2019

A072913 Numerators 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.

Original entry on oeis.org

1, 31, 3661, 76111, 58067611, 68165041, 187059457981, 3355156783231, 300222042894631, 327873266234371, 5194481903600608411, 5578681466128739761, 170044702211669500782121, 180514164422163370751221

Offset: 1

Vladeta Jovovic, Aug 10 2002

a(n) is also the numerator of binomial transform of (-1)^n/(n+1)^5

Jerry Metzger and Thomas Richards, A Prisoner Problem Variation, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.7.

Cf. A027459, A027462, A072914.

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)=numerator(1/4!*(x(n)^4+6*x(n)^2*y(n)+8*x(n)*z(n)+3*y(n)^2+6*w(n)))

Numerators 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))).

For n>=1, 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)=integral(x^(n-1)*(log(1-x))^4 dx, x=0..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).

A329108 First column of A027447.

Original entry on oeis.org

1, 7, 85, 415, 12019, 13489, 726301, 3144919, 30300391, 32160403, 4102360483, 4301068993, 758647585777, 788807993197, 817534859365, 3379894237615, 1007162177631295, 1036310871245335, 384227195120585215, 393975698131531915, 403382871919715515

Offset: 1

Sean A. Irvine, Nov 04 2019

nonn

This sequences was originally A027459, but differs from the current version of A027459 for n >= 14. This difference arises because the rational numbers in the matrix defining A027447 (and hence this sequence) are not reduced to the lowest common denominator, whereas the values in A027459 are.

Cf. A027447, A027459, A027460.

cp's OEIS Frontend

A027462 a(n) is the numerator of (-1/6) * Integral_{x=0..1} x^n * log^3(1-x).

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A072913 Numerators 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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A329108 First column of A027447.

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cp's OEIS Frontend

A027462 a(n) is the numerator of (-1/6) * Integral_{x=0..1} x^n * log^3(1-x).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A072913 Numerators 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

Author

Keywords

Comments

Links

Crossrefs

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

A329108 First column of A027447.

Views

Author

Keywords

Comments

Crossrefs

A072913 Numerators 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.