This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A027462 #39 Feb 27 2022 16:12:03
%S A027462 1,15,575,5845,874853,336581,129973303,1149858589,101622655189,
%T A027462 21945415349,31276937512951,33264031387717,77287019174361937,
%U A027462 81347802723340093,17055178843123409,142531324182321979
%N A027462 a(n) is the numerator of (-1/6) * Integral_{x=0..1} x^n * log^3(1-x).
%C A027462 Originally defined as the first column of A027448, but now contains numerator in reduced form (cf. A329122). - _Sean A. Irvine_, Nov 05 2019
%H A027462 Donal F. Connon, <a href="http://arxiv.org/abs/0710.4023">Some series and integrals involving the Riemann zeta function, binomial coefficients and the harmonic numbers. Vol. II(a)</a>, arXiv:0710.4023 [math.HO], 2007-2008.
%H A027462 Jerry Metzger and Thomas Richards, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Metzger/metz1.html">A Prisoner Problem Variation</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.7.
%F A027462 Numerators of sequence a(1,n) in (a(i,j))^4 where a(i,j) = 1/i if j <= i, 0 if j > i.
%F A027462 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
%F A027462 From _Groux Roland_, Feb 11 2011: (Start)
%F A027462 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.
%F A027462 For every k>=1 the first column of (a(i,j))^k is the binomial transform of (-1)^n/(n+1)^k.
%F A027462 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.
%F A027462 (End)
%t A027462 a[n_] := -1/6 Integrate[x^(n-1) Log[1-x]^3, {x, 0, 1}] // Numerator;
%t A027462 Table[a[n], {n, 1, 16}] (* _Jean-François Alcover_, Aug 06 2018 *)
%Y A027462 Cf. A027459.
%K A027462 nonn
%O A027462 1,2
%A A027462 _Olivier Gérard_
%E A027462 Corrected by _Vladeta Jovovic_, Aug 10 2002
%E A027462 Title changed by _Sean A. Irvine_, Nov 05 2019