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Decimal expansion of Product_{k >= 1} (k*(k+1))^(-1/(k*(k+1))), a constant related to the alternating Lüroth representations of real numbers.
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%I #13 Dec 15 2023 10:58:19

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

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

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

%N Decimal expansion of Product_{k >= 1} (k*(k+1))^(-1/(k*(k+1))), a constant related to the alternating Lüroth representations of real numbers.

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.8.1 Alternative representations [of real numbers], p. 62.

%H Sofia Kalpazidou, <a href="https://doi.org/10.1016/0022-314X(88)90099-6">Khintchine's constant for Lüroth representation</a>, Journal of Number Theory, Volume 29, Issue 2, June 1988, Pages 196-205.

%F Exp(-Sum_{n >= 1} (((1 + (-1)^(n+1))*Zeta(n+1) - 1)/n)). - After _Vaclav Kotesovec_'s formula for A244109.

%e 0.1292150184060998413415719000742197771573364620386787448773...

%t digits = 105; Exp[-NSum[((1 + (-1)^(n + 1))*Zeta[n + 1] - 1)/n, {n, 1, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> 2 digits, NSumTerms -> 200]] // RealDigits[#, 10, digits]& // First

%Y Cf. A131688, A244109, A245254.

%K nonn,cons

%O 0,2

%A _Jean-François Alcover_, Apr 24 2016

%E Offset corrected by _Andrey Zabolotskiy_, Dec 12 2023