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.
1, 31, 3661, 76111, 58067611, 68165041, 187059457981, 3355156783231, 300222042894631, 327873266234371, 5194481903600608411, 5578681466128739761, 170044702211669500782121, 180514164422163370751221
Offset: 1
Links
- Jerry Metzger and Thomas Richards, A Prisoner Problem Variation, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.7.
Programs
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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)))
Formula
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)
Extensions
More terms from Benoit Cloitre, Aug 13 2002
Comments