A397575 Decimal expansion of Integral_{0..1} Li_8(x)^2 dx, where Li_8 is the polylogarithm function.

3, 3, 5, 3, 5, 6, 7, 2, 3, 4, 7, 0, 4, 8, 4, 5, 7, 2, 2, 7, 4, 7, 1, 9, 4, 7, 6, 8, 0, 6, 9, 4, 2, 2, 6, 7, 0, 8, 7, 6, 0, 1, 3, 5, 5, 9, 6, 1, 0, 1, 8, 0, 2, 6, 2, 3, 4, 9, 4, 0, 0, 7, 4, 7, 9, 8, 4, 3, 7, 4, 7, 1, 0, 8, 2, 5, 9, 8, 2, 4, 2, 6, 0, 9, 1, 0, 0, 1, 5, 4, 0, 9, 6, 7, 0, 0, 7, 9, 0, 5, 0, 6, 8, 4, 1, 1

Offset

0,1

Links

Table of n, a(n) for n=0..105.

Pedro Freitas, Integrals of polylogarithmic functions, recurrence relations, and associated Euler sums, Math. Comput., Vol. 74, No. 251 (2005), pp. 1425-1440.

Eric Weisstein's World of Mathematics, Polylogarithm.

Formula

Equals 132*zeta(2)^2 - 264*zeta(3)*zeta(2) + 180*zeta(4)*zeta(2) - 96*zeta(5)*zeta(2) + 40*zeta(6)*zeta(2) - 12*zeta(7)*zeta(2) + 2*zeta(8)*zeta(2) - 6006*zeta(2) + 132*zeta(3)^2 + 62*zeta(4)^2 + 20*zeta(5)^2 + 6*zeta(6)^2 + 2*zeta(7)^2 + zeta(8)^2 - 4290*zeta(3) - 180*zeta(3)*zeta(4) + 2970*zeta(4) + 96*zeta(3)*zeta(5) - 68*zeta(4)*zeta(5) - 1650*zeta(5) - 40*zeta(3)*zeta(6) + 30*zeta(4)*zeta(6) - 20*zeta(5)*zeta(6) + 750*zeta(6) + 12*zeta(3)*zeta(7) - 10*zeta(4)*zeta(7) + 8*zeta(5)*zeta(7) - 6*zeta(6)*zeta(7) - 270*zeta(7) - 2*zeta(3)*zeta(8) + 2*zeta(4)*zeta(8) - 2*zeta(5)*zeta(8) + 2*zeta(6)*zeta(8) - 2*zeta(7)*zeta(8) + 70*zeta(8) - 10*zeta(9) + 12870.

Example

0.335356723470484572274719476806942267087601355961018026234940074798437...

Mathematica

RealDigits[12870 - 1001*Pi^2 + 110*Pi^4/3 + 71*Pi^6/63 + 209*Pi^8/9450 + 11*Pi^10/28350 + Pi^12/110250 + Pi^14/4465125 + Pi^16/89302500 - 4290*Zeta[3] - 44*Pi^2*Zeta[3] - 2*Pi^4*Zeta[3] - 8/189*Pi^6*Zeta[3] - Pi^8*Zeta[3]/4725 + 132*Zeta[3]^2 - 1650*Zeta[5] - 16*Pi^2*Zeta[5] - 34/45*Pi^4*Zeta[5] - 4/189*Pi^6*Zeta[5] - Pi^8*Zeta[5]/4725 + 96*Zeta[3]*Zeta[5] + 20*Zeta[5]^2 - 270*Zeta[7] - 2*Pi^2*Zeta[7] - 1/9*Pi^4*Zeta[7] - 2/315*Pi^6*Zeta[7] - Pi^8*Zeta[7]/4725 + 12*Zeta[3]*Zeta[7] + 8*Zeta[5]*Zeta[7] + 2*Zeta[7]^2 - 10*Zeta[9], 10, 110][[1]]
(* Alternative: *)
NIntegrate[PolyLog[8, x]^2, {x, 0, 1}, WorkingPrecision->110]

Crossrefs

Cf. A249651, A249652, A397550, A397555, A397576, A397577.

Sequence in context: A178984 A065507 A182731 * A267518 A101772 A233793

Adjacent sequences: A397567 A397568 A397574 * A397576 A397577 A397579

Keyword

nonn,cons

Author

Vaclav Kotesovec, Jul 01 2026

Status

approved