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a(n) is the numerator of (-1/6) * Integral_{x=0..1} x^n * log^3(1-x).
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%I #39 Feb 27 2022 16:12:03

%S 1,15,575,5845,874853,336581,129973303,1149858589,101622655189,

%T 21945415349,31276937512951,33264031387717,77287019174361937,

%U 81347802723340093,17055178843123409,142531324182321979

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

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

%H 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 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 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 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 From _Groux Roland_, Feb 11 2011: (Start)

%F 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 For every k>=1 the first column of (a(i,j))^k is the binomial transform of (-1)^n/(n+1)^k.

%F 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 (End)

%t a[n_] := -1/6 Integrate[x^(n-1) Log[1-x]^3, {x, 0, 1}] // Numerator;

%t Table[a[n], {n, 1, 16}] (* _Jean-François Alcover_, Aug 06 2018 *)

%Y Cf. A027459.

%K nonn

%O 1,2

%A _Olivier Gérard_

%E Corrected by _Vladeta Jovovic_, Aug 10 2002

%E Title changed by _Sean A. Irvine_, Nov 05 2019