A368211 Decimal expansion of Product_{k>=1} (1 - exp(-Pi*k/16)).

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

Offset

-2,2

Comments

In general, for 0 <= x < 1, QPochhammer(x) = (-8*x*QPochhammer(x^4)^8 + sqrt(QPochhammer(x^2)^24/QPochhammer(x^4)^8 + 64*x^2*QPochhammer(x^4)^16))^(1/8).

Links

Table of n, a(n) for n=-2..103.

Eric Weisstein's World of Mathematics, Dedekind Eta Function

Eric Weisstein's World of Mathematics, q-Pochhammer Symbol

Wikipedia, Dedekind eta function

Wikipedia, Euler function

Formula

Equals exp(Pi/384) * Gamma(1/4) / (((-16 + 6*sqrt(-24 + 22*sqrt(2)))/ (-8*(8 - 3*sqrt(-24 + 22*sqrt(2)))^2 + sqrt((-16 + 6*sqrt(-24 + 22*sqrt(2)))* (32*(-8 + 3*sqrt(-24 + 22*sqrt(2)))^3 + sqrt(2)*(99 + 70*sqrt(2))* (-136 + 96*sqrt(2) + 3*sqrt(2792 - 1984*sqrt(2) + sqrt(-1201560 + 849766*sqrt(2))))^3))))^(1/8) * (2*Pi)^(3/4)).

Equals (-8*exp(-Pi/16)*A292863^8 + sqrt(A292862^24 / A292863^8 + 64*exp(-Pi/8) * A292863^16))^(1/8).

Example

0.001311568896679258886414738411968760891094304411218465289631108459577842...

Mathematica

RealDigits[QPochhammer[E^(-Pi/16)], 10, 120][[1]]
RealDigits[E^(Pi/384) * Gamma[1/4] / (((-16 + 6*Sqrt[-24 + 22*Sqrt[2]])/ (-8*(8 - 3*Sqrt[-24 + 22*Sqrt[2]])^2 + Sqrt[(-16 + 6*Sqrt[-24 + 22*Sqrt[2]])* (32*(-8 + 3*Sqrt[-24 + 22*Sqrt[2]])^3 + Sqrt[2]*(99 + 70*Sqrt[2])* (-136 + 96*Sqrt[2] + 3*Sqrt[2792 - 1984*Sqrt[2] + Sqrt[-1201560 + 849766*Sqrt[2]]])^3)]))^(1/8) * (2*Pi)^(3/4)), 10, 120][[1]]

Crossrefs

Cf. A292862, A292863, A259147, A259148, A259149.

Sequence in context: A294589 A204027 A080842 * A216948 A183944 A145661

Adjacent sequences: A368208 A368209 A368210 * A368212 A368213 A368214

Keyword

nonn,cons

Author

Vaclav Kotesovec, Dec 17 2023

Status

approved