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A363179 Decimal expansion of Product_{k>=1} (1 - exp(-15*Pi*k)). 14
9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 6, 5, 7, 7, 4, 1, 1, 4, 5, 5, 8, 7, 8, 7, 5, 9, 1, 3, 2, 1, 9, 2, 0, 8, 5, 4, 4, 7, 3, 4, 8, 9, 1, 0, 6, 1, 9, 1, 4, 0, 0, 1, 3, 9, 9, 8, 5, 6, 2, 8, 4, 4, 1, 8, 9, 2, 9, 8, 6, 8, 0, 6, 4, 2, 7, 6, 6, 1, 1, 7, 3, 6, 6, 7, 5, 6, 5, 5, 0, 1, 5, 3, 8, 1, 7, 8 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
LINKS
Wikipedia, Theta function.
FORMULA
Equals phi(exp(-30*Pi))^(5/2) / (phi(exp(-60*Pi)) * theta_3(0, exp(-15*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(5*Pi/8) * Gamma(1/4) * (2 - sqrt(3))^(55/24) * (2 + sqrt(3))^(13/12) * (sqrt(5) - 2)^(5/4) * (3 + sqrt(5)) * (2 + sqrt(2)*3^(3/4)*5^(1/4) + sqrt(2)*15^(1/4))^(3/2) * (-15^(1/4) + sqrt(4 + sqrt(15)))^5 * ((15^(1/4) + sqrt(4 + sqrt(15)))^(5/2) / (Pi^(3/4) * 2048 * 3^(3/8) * sqrt(5) * (2*(7 + 3*sqrt(3) + sqrt(5) + sqrt(2)*3^(1/4)*5^(3/4) + sqrt(2)*15^(1/4) + sqrt(15)))^(1/4) * (((2 + sqrt(3))^4 * (1 + sqrt(5))^12 * (15^(1/4) + sqrt(4 + sqrt(15)))^12) / 16777216 - sqrt(-1 + ((2 + sqrt(3))^8 * (1 + sqrt(5))^24 * (15^(1/4) + sqrt(4 + sqrt(15)))^24) / 281474976710656))^(1/8))).
EXAMPLE
0.99999999999999999999657741145587875913219208544734891061914001399856284...
MATHEMATICA
RealDigits[QPochhammer[E^(-15*Pi)], 10, 120][[1]]
RealDigits[QPochhammer[E^(-30*Pi)]^(5/2) / QPochhammer[E^(-60*Pi)] / EllipticTheta[3, 0, Exp[-15*Pi]]^(1/2), 10, 120][[1]]
RealDigits[E^(5*Pi/8) * Gamma[1/4] * (2 - Sqrt[3])^(55/24) * (2 + Sqrt[3])^(13/12) * (Sqrt[5] - 2)^(5/4) * (3 + Sqrt[5]) * (2 + Sqrt[2]*3^(3/4)*5^(1/4) + Sqrt[2]*15^(1/4))^(3/2) * (-15^(1/4) + Sqrt[4 + Sqrt[15]])^5 * ((15^(1/4) + Sqrt[4 + Sqrt[15]])^(5/2) / (Pi^(3/4) * 2048 * 3^(3/8) * Sqrt[5] * (2*(7 + 3*Sqrt[3] + Sqrt[5] + Sqrt[2]*3^(1/4)*5^(3/4) + Sqrt[2]*15^(1/4) + Sqrt[15]))^(1/4) * (((2 + Sqrt[3])^4 * (1 + Sqrt[5])^12 * (15^(1/4) + Sqrt[4 + Sqrt[15]])^12) / 16777216 - Sqrt[-1 + ((2 + Sqrt[3])^8 * (1 + Sqrt[5])^24 * (15^(1/4) + Sqrt[4 + Sqrt[15]])^24) / 281474976710656])^(1/8))), 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)), A363178 phi(exp(-13*Pi)), A363119 phi(exp(-14*Pi)), A292864 phi(exp(-16*Pi)), A363120 phi(exp(-18*Pi)), A363021 phi(exp(-20*Pi)).
Sequence in context: A363020 A363178 A363119 * A292864 A363120 A363021
KEYWORD
nonn,cons
AUTHOR
Vaclav Kotesovec, May 19 2023
STATUS
approved

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Last modified February 23 11:40 EST 2024. Contains 370283 sequences. (Running on oeis4.)