A259928 Decimal expansion of the infinite double sum S = Sum_{m>=1} (Sum_{n>=1} 1/(m^2*n*(m+n)^3)).

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

Offset: 0

Author

Jean-François Alcover, Jul 09 2015

Keywords

• nonn
• cons
• easy

Examples

			0.16955717699740818995241965496515342131696958167214226030706811...
		

Links

• StackExchange, Integral of polylogarithms and logs in closed form
• Eric Weisstein's MathWorld, Polygamma Function.
• Wikipedia, Polygamma Function.

Crossrefs

Cf. A258987, A258989, A259927.

Programs

Mathematica

RealDigits[Pi^6/5670, 10, 103] // First

PARI

Pi^6/5670 \\ Michel Marcus, Jul 09 2015

Formula

S = (7/4)*zeta(6) - zeta(3)^2/2 - sum_{m>=1} (PolyGamma(1, m+1)/m^4) + (1/2)*sum_{m>=1} (PolyGamma(2, m+1)/m^3), where sum_{m>=1} (PolyGamma(1, m+1)/m^4) is A258989, the second sum being A259927.

S simplifies to zeta(6)/6 = Pi^6/5670.

2*A258987 + 6*S = zeta(3)^2.