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%I #36 Feb 13 2024 05:13:15
%S 0,0,4,18,164,2600,64072,2272032,109735488,6930012672,554528623104,
%T 54840436992000,6568892183808000,937223951339520000,
%U 157057344897601536000,30545188599606047539200,6823697557721234964480000,1735362552287102663393280000
%N a(n) = (n!)^2 * Sum_{k=1..n-1} 1/(k*(n-k))^2.
%H Seiichi Manyama, <a href="/A302827/b302827.txt">Table of n, a(n) for n = 0..250</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Polylogarithm.html">Polylogarithm</a>.
%F Recurrence: n*(2*n - 3)*a(n) = (n-1)*(6*n^3 - 25*n^2 + 33*n - 12)*a(n-1) - (n-2)^3*(6*n^3 - 29*n^2 + 42*n - 15)*a(n-2) + (n-3)^4*(n-2)^3*(2*n - 1)*a(n-3).
%F a(n) / (n!)^2 ~ Pi^2/(3*n^2) + 4*log(n)/n^3.
%p seq(factorial(n)^2*add(1/(k*(n-k))^2,k=1..n-1),n=0..20); # _Muniru A Asiru_, May 16 2018
%t Table[n!^2*Sum[1/(k*(n-k))^2, {k, 1, n-1}], {n, 0, 20}]
%t CoefficientList[Series[PolyLog[2, x]^2, {x, 0, 20}], x] * Range[0,20]!^2
%o (GAP) List([0..20],n->Factorial(n)^2*Sum([1..n-1],k->1/(k*(n-k))^2)); # _Muniru A Asiru_, May 16 2018
%Y Cf. A052517, A304581, A304582, A304589, A304654, A370225, A370226.
%K nonn
%O 0,3
%A _Vaclav Kotesovec_, May 15 2018
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