This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A370225 #9 Feb 16 2025 08:34:06
%S A370225 0,0,0,36,432,7500,191700,6899592,334858944,21143500992,1687177771200,
%T A370225 166164785309952,19807714583407872,2811642170175940608,
%U A370225 468723024212650002432,90691236890980830683136,20158684374653040962764800,5101828835401918167928012800,1458612909355515900114154291200
%N A370225 a(n) = n!^2 * [x^n] polylog(2,x)^3.
%H A370225 Vaclav Kotesovec, <a href="/A370225/b370225.txt">Table of n, a(n) for n = 0..250</a>
%H A370225 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Polylogarithm.html">Polylogarithm</a>.
%H A370225 Wikipedia, <a href="http://en.wikipedia.org/wiki/Polylogarithm">Polylogarithm</a>.
%F A370225 Recurrence: (n-1) * n^2 * (24*n^9 - 648*n^8 + 7928*n^7 - 58928*n^6 + 297426*n^5 - 1054693*n^4 + 2579632*n^3 - 4091937*n^2 + 3732024*n - 1463688) * a(n) = 4*(n-1)^2 * (36*n^12 - 1146*n^11 + 16806*n^10 - 151862*n^9 + 950049*n^8 - 4345179*n^7 + 14812504*n^6 - 37460061*n^5 + 68666655*n^4 - 87462969*n^3 + 72065619*n^2 - 33585192*n + 6382800) * a(n-1) - (n-2)^2 * (360*n^14 - 13560*n^13 + 237600*n^12 - 2585176*n^11 + 19644502*n^10 - 110806171*n^9 + 478897261*n^8 - 1605269146*n^7 + 4161296764*n^6 - 8214107379*n^5 + 12005416617*n^4 - 12428493288*n^3 + 8478515088*n^2 - 3337578000*n + 550726128) * a(n-2) + 8*(n-3)^4 * (60*n^14 - 2370*n^13 + 43430*n^12 - 492632*n^11 + 3892949*n^10 - 22819863*n^9 + 102674094*n^8 - 359603394*n^7 + 978009142*n^6 - 2031308188*n^5 + 3125785870*n^4 - 3399739356*n^3 + 2424585966*n^2 - 989615976*n + 166980888) * a(n-3) - (n-4)^4 * (n-3)^2 * (360*n^14 - 14880*n^13 + 284280*n^12 - 3348872*n^11 + 27395534*n^10 - 165978041*n^9 + 772288075*n^8 - 2803517230*n^7 + 7923458764*n^6 - 17127191793*n^5 + 27415542063*n^4 - 30937263696*n^3 + 22788329436*n^2 - 9544987584*n + 1636611264) * a(n-4) + 4*(n-5)^5 * (n-4)^2 * (n-3)^2 * (36*n^13 - 1374*n^12 + 24000*n^11 - 256474*n^10 + 1894307*n^9 - 10337820*n^8 + 43164880*n^7 - 139027139*n^6 + 340176690*n^5 - 610956909*n^4 + 764185680*n^3 - 614613168*n^2 + 276176448*n - 49654512) * a(n-5) - (n-6)^6 * (n-5)^5 * (n-4)^2 * (n-3)^2 * (24*n^9 - 432*n^8 + 3608*n^7 - 19560*n^6 + 77082*n^5 - 216339*n^4 + 399768*n^3 - 441651*n^2 + 252364*n - 52860) * a(n-6).
%F A370225 a(n)/(n!)^2 ~ Pi^4 / (12*n^2) * (1 + 24*log(n)/(Pi^2*n)).
%t A370225 CoefficientList[Series[PolyLog[2, x]^3, {x, 0, 20}], x] * Range[0, 20]!^2
%t A370225 Table[n!^2 * Sum[Sum[1/(k*(j-k))^2, {k, 1, j-1}] * 1/(n-j)^2, {j, 1, n-1}], {n, 0, 20}]
%Y A370225 Cf. A302827, A370226.
%K A370225 nonn
%O A370225 0,4
%A A370225 _Vaclav Kotesovec_, Feb 12 2024