OFFSET
0,3
COMMENTS
In general, for m > 1, Sum_{k=1..n-1} 1/(k*(n-k))^m is asymptotic to 2*Zeta(m)/n^m.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..150
Eric Weisstein's World of Mathematics, Polylogarithm.
FORMULA
Recurrence: n^2*(12*n^4 - 108*n^3 + 354*n^2 - 501*n + 260)*a(n) = 2*(n-1)^2*(24*n^7 - 306*n^6 + 1620*n^5 - 4599*n^4 + 7516*n^3 - 7015*n^2 + 3444*n - 696)*a(n-1) - 6*(n-2)^5*(12*n^7 - 162*n^6 + 906*n^5 - 2700*n^4 + 4583*n^3 - 4378*n^2 + 2163*n - 436)*a(n-2) + 2*(n-3)^5*(n-2)^3*(24*n^7 - 342*n^6 + 2004*n^5 - 6201*n^4 + 10816*n^3 - 10497*n^2 + 5208*n - 1048)*a(n-3) - (n-4)^6*(n-3)^5*(n-2)^3*(12*n^4 - 60*n^3 + 102*n^2 - 69*n + 17)*a(n-4).
a(n) / (n!)^3 ~ 2*Zeta(3)/n^3.
MAPLE
seq(factorial(n)^3*add(1/(k*(n-k))^3, k=1..n-1), n=0..20); # Muniru A Asiru, May 16 2018
MATHEMATICA
Table[n!^3*Sum[1/(k*(n-k))^3, {k, 1, n-1}], {n, 0, 20}]
CoefficientList[Series[PolyLog[3, x]^2, {x, 0, 20}], x] * Range[0, 20]!^3
PROG
(GAP) List([0..20], n->Factorial(n)^3*Sum([1..n-1], k->1/(k*(n-k))^3)); # Muniru A Asiru, May 16 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, May 15 2018
STATUS
approved