A302827 a(n) = (n!)^2 * Sum_{k=1..n-1} 1/(k*(n-k))^2.

0, 0, 4, 18, 164, 2600, 64072, 2272032, 109735488, 6930012672, 554528623104, 54840436992000, 6568892183808000, 937223951339520000, 157057344897601536000, 30545188599606047539200, 6823697557721234964480000, 1735362552287102663393280000

Offset: 0

Author

Vaclav Kotesovec, May 15 2018

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..250
• Eric Weisstein's World of Mathematics, Polylogarithm.

Crossrefs

Cf. A052517, A304581, A304582, A304589, A304654, A370225, A370226.

Programs

GAP

List([0..20],n->Factorial(n)^2*Sum([1..n-1],k->1/(k*(n-k))^2)); # Muniru A Asiru, May 16 2018

Maple

seq(factorial(n)^2*add(1/(k*(n-k))^2,k=1..n-1),n=0..20); # Muniru A Asiru, May 16 2018

Mathematica

Table[n!^2*Sum[1/(k*(n-k))^2, {k, 1, n-1}], {n, 0, 20}]
CoefficientList[Series[PolyLog[2, x]^2, {x, 0, 20}], x] * Range[0,20]!^2

Formula

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).

a(n) / (n!)^2 ~ Pi^2/(3*n^2) + 4*log(n)/n^3.