cp's OEIS Frontend

A304589 a(n) = (n!)^3 * Sum_{k=1..n-1} 1/(k*(n-k))^3.

%I A304589 #15 Feb 16 2025 08:33:54
%S A304589 0,0,8,54,1240,70000,7941968,1589632128,512918521344,249820864339968,
%T A304589 174720109813751808,168721560082538496000,217977447876560510976000,
%U A304589 367117517435096337481728000,788739873984137255456342016000,2122296978948474538763602624512000
%N A304589 a(n) = (n!)^3 * Sum_{k=1..n-1} 1/(k*(n-k))^3.
%C A304589 In general, for m > 1, Sum_{k=1..n-1} 1/(k*(n-k))^m is asymptotic to 2*Zeta(m)/n^m.
%H A304589 Seiichi Manyama, <a href="/A304589/b304589.txt">Table of n, a(n) for n = 0..150</a>
%H A304589 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Polylogarithm.html">Polylogarithm</a>.
%F A304589 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).
%F A304589 a(n) / (n!)^3 ~ 2*Zeta(3)/n^3.
%p A304589 seq(factorial(n)^3*add(1/(k*(n-k))^3,k=1..n-1),n=0..20); # _Muniru A Asiru_, May 16 2018
%t A304589 Table[n!^3*Sum[1/(k*(n-k))^3, {k, 1, n-1}], {n, 0, 20}]
%t A304589 CoefficientList[Series[PolyLog[3, x]^2, {x, 0, 20}], x] * Range[0,20]!^3
%o A304589 (GAP) List([0..20],n->Factorial(n)^3*Sum([1..n-1],k->1/(k*(n-k))^3)); # _Muniru A Asiru_, May 16 2018
%Y A304589 Cf. A052517, A302827, A002547, A002548, A304581, A304582, A304655.
%K A304589 nonn
%O A304589 0,3
%A A304589 _Vaclav Kotesovec_, May 15 2018