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%I #19 Jul 11 2015 17:30:41
%S 1,31,3661,76111,58067611,68165041,187059457981,3355156783231,
%T 300222042894631,327873266234371,5194481903600608411,
%U 5578681466128739761,170044702211669500782121,180514164422163370751221
%N 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.
%C a(n) is also the numerator of binomial transform of (-1)^n/(n+1)^5
%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 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))).
%F 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)
%o (PARI) 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)))
%Y Cf. A027459, A027462, A072914.
%K easy,nonn,frac
%O 1,2
%A _Vladeta Jovovic_, Aug 10 2002
%E More terms from _Benoit Cloitre_, Aug 13 2002
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