A397576 Decimal expansion of Integral_{0..1} Li_6(x)^2 dx, where Li_6 is the polylogarithm function.

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

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 14*zeta(2)^2 - 28*zeta(3)*zeta(2) + 18*zeta(4)*zeta(2) - 8*zeta(5)*zeta(2) + 2*zeta(6)*zeta(2) - 420*zeta(2) + 14*zeta(3)^2 + 6*zeta(4)^2 + 2*zeta(5)^2 + zeta(6)^2 - 336*zeta(3) - 18*zeta(3)*zeta(4) + 224*zeta(4) + 8*zeta(3)*zeta(5) - 6*zeta(4)*zeta(5) - 112*zeta(5) - 2*zeta(3)*zeta(6) + 2*zeta(4)*zeta(6) - 2*zeta(5)*zeta(6) + 40*zeta(6) - 8*zeta(7) + 924.

Example

0.341860754469721529107960123637445903641137940143135312301900721227326...

Mathematica

RealDigits[924 - 70*Pi^2 + 259*Pi^4/90 + 143*Pi^6/1890 + 31*Pi^8/28350 + Pi^10/42525 + Pi^12/893025 - 336*Zeta[3] - 14/3*Pi^2*Zeta[3] - 1/5*Pi^4*Zeta[3] - 2/945*Pi^6*Zeta[3] + 14*Zeta[3]^2 - 112*Zeta[5] - 4/3*Pi^2*Zeta[5] - 1/15*Pi^4*Zeta[5] - 2/945*Pi^6*Zeta[5] + 8*Zeta[3]*Zeta[5] + 2*Zeta[5]^2 - 8*Zeta[7], 10, 110][[1]]
(* Alternative: *)
NIntegrate[PolyLog[6, x]^2, {x, 0, 1}, WorkingPrecision->110]

Crossrefs

Cf. A249651, A249652, A397550, A397555, A397577, A397575.

Sequence in context: A326503 A378913 A105578 * A378227 A198739 A158076

Adjacent sequences: A397571 A397574 A397575 * A397577 A397579 A397580

Keyword

nonn,cons

Author

Vaclav Kotesovec, Jul 01 2026

Status

approved