A series involving harmonic numbers

Does anyone know the exact value of this:
k=1(1)kHkk
or this:
k=1(1)kH(2)kk
Thanks!


Thanks again for the answers! I found very interesting that the integral gives exact values up to r=3 but from 4 this integral gives not exact values:

01Li4(t)t(1t)dt

because wolfram says that “no results found in terms of standard mathematical functions”

Answer

From my answer in my previous avatar, denoting
A(p,q)=k=1(1)k+1H(p)kkq
we have
A(1,1)=ζ(2)log2(2)2
and
A(2,1)=ζ(3)ζ(2)log(2)2

Attribution
Source : Link , Question Author : Antal Korpa , Answer Author : Community

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