A398908 - OEIS

A398908

Decimal expansion of Sum_{i,j,k>=0} (-1)^(i+j+k) * i!*j!*k!/(i+j+k+2)!.

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

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OFFSET

0,1

COMMENTS

Sum_{i,j,k>=0} x^(i+j+k) * i!*j!*k!/(i+j+k+2)! = 6 * (zeta(2)/2 + log(1-x)*log(2-x) + Li_2(-(1-x))) / ((3-x) * x^2) for abs(x) < 1.

LINKS

Table of n, a(n) for n=0..104.

Cornel Ioan Vălean, More (Almost) Impossible Integrals, Sums, and Series, Springer Cham, 2023. See section 4.60, p. 456, section 5.58, p. 469, section 6.60, pp. 807-811.

FORMULA

Equals 3*log(2)*log(3)/2 + 3*zeta(2)/4 + 3*Li_2(-2)/2.

Equals 3*log(2)*log(3)/2 - 3*log(2)^2/4 - 3*zeta(2)/4 - 3*Li_2(-1/2)/2.

EXAMPLE

0.220831015438861887453642414398899689002730207451334...

MATHEMATICA

RealDigits[3*Log[2]*Log[3]/2 + 3*Zeta[2]/4 + 3*PolyLog[2, -2]/2, 10, 120][[1]]

PROG

(PARI) 3*log(2)*log(3)/2 + 3*zeta(2)/4 + 3*polylog(2, -2)/2

CROSSREFS

Cf. A013661 (Sum_{i,j>=0} i!*j!/(i+j+2)!), A091476 (Sum_{i,j,k>=0} i!*j!*k!/(i+j+k+2)!).

Cf. A002162, A002391, A266576, A355234.