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%I #12 Oct 27 2023 13:10:07
%S 1,0,0,0,0,0,0,1,5,0,7,0,1,7,5,0,2,5,0,0,2,3,9,8,9,4,9,3,8,6,9,8,7,1,
%T 4,6,7,9,7,3,7,6,1,0,0,6,4,3,0,5,0,7,4,0,5,6,9,0,1,9,9,9,8,8,5,2,0,8,
%U 8,7,1,3,4,4,2,6,9,4,9,7,1,7,6,1,8,7,2,8,7,4,6,7,3,2,5,8,5,1,0,0,2,8,5,0,4
%N Decimal expansion of Product_{k>=1} (1 + exp(-5*Pi*k)).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DedekindEtaFunction.html">Dedekind Eta Function</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/q-PochhammerSymbol.html">q-Pochhammer Symbol</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Dedekind_eta_function">Dedekind eta function</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Euler_function">Euler function</a>
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F Root r of the equation 2^(3/4)*r^6 + 2^(17/8)*exp(5*Pi/24)*r^5 + 2^(5/8)*exp(25*Pi/24)*r - exp(5*Pi/4) = 0.
%F Equals exp(5*Pi/24) * sqrt(2 + sqrt(5) - sqrt((15 + 7*sqrt(5))/2))/2^(1/8). - _Vaclav Kotesovec_, May 13 2023
%e 1.000000150701750250023989493869871467973761006430507405690199988520887...
%t RealDigits[r/.FindRoot[2^(3/4)*r^6 + 2^(17/8)*E^(5*Pi/24)*r^5 + 2^(5/8)*E^(25*Pi/24)*r - E^(5*Pi/4) == 0, {r, 1}, WorkingPrecision -> 120], 10, 120][[1]]
%t RealDigits[QPochhammer[-1, E^(-5*Pi)]/2, 10, 120][[1]]
%t RealDigits[Exp[5*Pi/24]*Sqrt[2 + Sqrt[5] - Sqrt[(15 + 7*Sqrt[5])/2]]/2^(1/8), 10, 120][[1]] (* _Vaclav Kotesovec_, May 13 2023 *)
%o (PARI) polrootsreal(2^(3/4)*'x^6 + 2^(17/8)*exp(5*Pi/24)*'x^5 + 2^(5/8)*exp(25*Pi/24)*'x - exp(5*Pi/4))[2] \\ _Charles R Greathouse IV_, Mar 04 2018
%Y Cf. A292819, A292820, A292821, A292887, A292822, A292905.
%K nonn,cons
%O 1,9
%A _Vaclav Kotesovec_, Sep 26 2017
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