A370226 a(n) = n!^2 * [x^n] polylog(2,x)^4.

0, 0, 0, 0, 576, 14400, 424800, 16405200, 827179584, 53370793728, 4311612000000, 427527300499200, 51134102684222976, 7266620131443459072, 1211052516384021083136, 234033301581064751001600, 51924413277653839769124864, 13111663349134716037934874624, 3739245464888523341104099885056

Offset: 0

Author

Vaclav Kotesovec, Feb 12 2024

Keywords

• nonn

Comments

In general, for m >= 1, [x^n] polylog(2,x)^m ~ m*zeta(2)^(m-1)/n^2 = m * Pi^(2*m-2) / (6^(m-1) * n^2).

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..250
• Vaclav Kotesovec, Recurrence (of order 10)
• Eric Weisstein's World of Mathematics, Polylogarithm.
• Wikipedia, Polylogarithm.

Crossrefs

Cf. A302827, A370225.

Programs

Mathematica

CoefficientList[Series[PolyLog[2, x]^4, {x, 0, 20}], x] * Range[0, 20]!^2
Table[n!^2 * Sum[Sum[1/(k*(j-k))^2, {k, 1, j-1}] * Sum[1/(k*(n-j-k))^2, {k, 1, n-j-1}], {j, 1, n-1}], {n, 0, 20}]

Formula

a(n)/(n!)^2 ~ Pi^6 / (54*n^2).