A363178 Decimal expansion of Product_{k>=1} (1 - exp(-13*Pi*k)).

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

Offset

0,1

Links

Table of n, a(n) for n=0..105.

Mathematics Stack Exchange, Additional values of Dedekind's eta function in radical form.

Wikipedia, Theta function.

Formula

Equals phi(exp(-26*Pi))^(5/2) / (phi(exp(-52*Pi)) * theta_3(0, exp(-13*Pi))^(1/2)), where phi(q) = Product_{k>=1} (1 - q^k) is the Euler modular function and theta_3 is the 3rd Jacobi theta function.

Equals exp(13*Pi/24) * Gamma(1/4) * ((-(11 - 6*sqrt(3))^(1/6) + (11 + 6*sqrt(3))^(1/6)) / ((11 - 6*sqrt(3))^(1/6) * (1 + 2*sqrt(3)) + (-1 + 2*sqrt(3)) * (11 + 6*sqrt(3))^(1/6)))^(1/4) * (-5 - (-91 + 39*sqrt(3) - 18*sqrt(13) + 15*sqrt(39))^(1/3) + (91 + 39*sqrt(3) + 18*sqrt(13) + 15*sqrt(39))^(1/3))^(3/4) * (sqrt(sqrt(14 + (13*(821 - 72*sqrt(3)))^(1/3) + (13*(821 + 72*sqrt(3)))^(1/3)) + sqrt(26 + (13*(821 - 72*sqrt(3)))^(1/3) + (13*(821 + 72*sqrt(3)))^(1/3))) / (Pi^(3/4) * 3 * 2^(11/8) * sqrt(13) * ((1/2985984)*(sqrt(14 + (13*(821 - 72*sqrt(3)))^(1/3) + (13*(821 + 72*sqrt(3)))^(1/3)) + sqrt(26 + (13*(821 - 72*sqrt(3)))^(1/3) + (13*(821 + 72*sqrt(3)))^(1/3)))^12 - sqrt(-1 + (1/8916100448256)*(sqrt(14 + (13*(821 - 72*sqrt(3)))^(1/3) + (13*(821 + 72*sqrt(3)))^(1/3)) + sqrt(26 + (13*(821 - 72*sqrt(3)))^(1/3) + (13*(821 + 72*sqrt(3)))^(1/3)))^24))^(1/8))).

Example

0.99999999999999999816723239432842242817700113854738989073221955396667771...

Mathematica

RealDigits[QPochhammer[E^(-13*Pi)], 10, 120][[1]]
RealDigits[QPochhammer[E^(-26*Pi)]^(5/2) / QPochhammer[E^(-52*Pi)] / EllipticTheta[3, 0, Exp[-13*Pi]]^(1/2), 10, 120][[1]]
RealDigits[E^(13*Pi/24) * Gamma[1/4] * ((-(11 - 6*Sqrt[3])^(1/6) + (11 + 6*Sqrt[3])^(1/6)) / ((11 - 6*Sqrt[3])^(1/6) * (1 + 2*Sqrt[3]) + (-1 + 2*Sqrt[3]) * (11 + 6*Sqrt[3])^(1/6)))^(1/4) * (-5 - (-91 + 39*Sqrt[3] - 18*Sqrt[13] + 15*Sqrt[39])^(1/3) + (91 + 39*Sqrt[3] + 18*Sqrt[13] + 15*Sqrt[39])^(1/3))^(3/4) * (Sqrt[Sqrt[14 + (13*(821 - 72*Sqrt[3]))^(1/3) + (13*(821 + 72*Sqrt[3]))^(1/3)] + Sqrt[26 + (13*(821 - 72*Sqrt[3]))^(1/3) + (13*(821 + 72*Sqrt[3]))^(1/3)]] / (Pi^(3/4) * 3 * 2^(11/8) * Sqrt[13] * ((1/2985984)*(Sqrt[14 + (13*(821 - 72*Sqrt[3]))^(1/3) + (13*(821 + 72*Sqrt[3]))^(1/3)] + Sqrt[26 + (13*(821 - 72*Sqrt[3]))^(1/3) + (13*(821 + 72*Sqrt[3]))^(1/3)])^12 - Sqrt[-1 + (1/8916100448256)*(Sqrt[14 + (13*(821 - 72*Sqrt[3]))^(1/3) + (13*(821 + 72*Sqrt[3]))^(1/3)] + Sqrt[26 + (13*(821 - 72*Sqrt[3]))^(1/3) + (13*(821 + 72*Sqrt[3]))^(1/3)])^24])^(1/8))), 10, 120][[1]]
RealDigits[E^(13*Pi/24) * Gamma[1/4] / (Pi^(3/4) * 2^(7/8) * Sqrt[13*Root[1 + 22*#1 + 224*#1^2 + 1366*#1^3 + 5456*#1^4 + 14758*#1^5 + 27158*#1^6 + 33094*#1^7 + 23936*#1^8 + 8854*#1^9 + 7952*#1^10 - 22058*#1^11 + #1^12 & , 1]]), 10, 120][[1]]

Crossrefs

Cf. A259148 phi(exp(-Pi)), A259149 phi(exp(-2*Pi)), A292888 phi(exp(-3*Pi)), A259150 phi(exp(-4*Pi)), A292905 phi(exp(-5*Pi)), A363018 phi(exp(-6*Pi)), A363117 phi(exp(-7*Pi)), A259151 phi(exp(-8*Pi)), A363118 phi(exp(-9*Pi)), A363019 phi(exp(-10*Pi)), A363081 phi(exp(-11*Pi)), A363020 phi(exp(-12*Pi)), A363119 phi(exp(-14*Pi)), A363179 phi(exp(-15*Pi)), A292864 phi(exp(-16*Pi)), A363120 phi(exp(-18*Pi)), A363021 phi(exp(-20*Pi)).

Sequence in context: A363019 A363081 A363020 * A363119 A363179 A292864

Adjacent sequences: A363175 A363176 A363177 * A363179 A363180 A363181

Keyword

nonn,cons

Author

Vaclav Kotesovec, May 19 2023

Status

approved