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

1,2

Comments

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

Links

Table of n, a(n) for n=1..16.

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.

Formula

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)

Mathematica

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 *)

Crossrefs

Cf. A027459.

Sequence in context: A012178 A012229 A212931 * A329122 A027534 A001236

Adjacent sequences: A027459 A027460 A027461 * A027463 A027464 A027465

Keyword

nonn

Author

Olivier Gérard

Extensions

Corrected by Vladeta Jovovic, Aug 10 2002

Title changed by Sean A. Irvine, Nov 05 2019

Status

approved