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Numerator of Sum_{k=1..n-1} 1/(k*(n-k))^2.
3

%I #10 May 15 2018 11:10:14

%S 0,0,1,1,41,13,8009,161,190513,167101,13371157,21857,316786853,371449,

%T 52598187029,260957190289,129548894873,3562512061,295728132584141,

%U 814542451061,105590441859671453,21013691164284241,2988054680665783,5623939943287,1567371864703176307

%N Numerator of Sum_{k=1..n-1} 1/(k*(n-k))^2.

%C Sum_{k=1..n-1} 1/(k*(n-k))^2 is asymptotic to Pi^2/(3*n^2) + 4*log(n)/n^3.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Dilogarithm.html">Dilogarithm</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Polylogarithm.html">Polylogarithm</a>.

%e 0, 0, 1, 1/2, 41/144, 13/72, 8009/64800, 161/1800, 190513/2822400, ...

%t CoefficientList[Series[PolyLog[2, x]^2, {x, 0, 25}], x]//Numerator

%t Table[Sum[1/(k*(n - k))^2, {k, 1, n - 1}], {n, 0, 25}]//Numerator

%Y Cf. A002547, A304582, A302827.

%K nonn,frac

%O 0,5

%A _Vaclav Kotesovec_, May 15 2018