A259150 -id:A259150 - cp's OEIS frontend

A363117 Decimal expansion of Product_{k>=1} (1 - exp(-7Pik)).

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

Offset: 0

Vaclav Kotesovec, May 15 2023

			0.999999999718573154172243658382901236462919560257076490298122086100117...

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)), 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)), A363179 phi(exp(-15*Pi)), A292864 phi(exp(-16*Pi)), A363120 phi(exp(-18*Pi)), A363021 phi(exp(-20*Pi)).

RealDigits[E^(7*Pi/24) * Gamma[1/4] * ((Sqrt[5 - Sqrt[7]] - Sqrt[3*Sqrt[7] - 7]) * (2^(1/4) * Sqrt[5 + Sqrt[7]] + (56 + 23*Sqrt[7])^(1/4)))^(1/4) / (2^(19/16) * 7^(7/16) * Pi^(3/4)), 10, 120][[1]]
RealDigits[QPochhammer[E^(-7*Pi)], 10, 120][[1]]

Equals exp(7*Pi/24) * Gamma(1/4) * ((sqrt(5 - sqrt(7)) - sqrt(3*sqrt(7) - 7)) * (2^(1/4) * sqrt(5 + sqrt(7)) + (56 + 23*sqrt(7))^(1/4)))^(1/4) / (2^(19/16) * 7^(7/16) * Pi^(3/4)).

A363118 Decimal expansion of Product_{k>=1} (1 - exp(-9Pik)).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, May 15 2023

			0.999999999999474451482399078943233394928797164400527513438819873918260...

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)), 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)), A363179 phi(exp(-15*Pi)), A292864 phi(exp(-16*Pi)), A363120 phi(exp(-18*Pi)), A363021 phi(exp(-20*Pi)).

RealDigits[E^(3*Pi/8) * Gamma[1/4] * ((3*(6 + 7*Sqrt[3] + 3*Sqrt[14*Sqrt[3] - 15]))^(1/3) - 3)^(1/3) / (3 * 2^(7/8) * Pi^(3/4)), 10, 120][[1]]
RealDigits[QPochhammer[E^(-9*Pi)], 10, 120][[1]]

Equals exp(3*Pi/8) * Gamma(1/4) * ((3*(6 + 7*sqrt(3) + 3*sqrt(14*sqrt(3) - 15)))^(1/3) - 3)^(1/3) / (3 * 2^(7/8) * Pi^(3/4)).

A363119 Decimal expansion of Product_{k>=1} (1 - exp(-14Pik)).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, May 15 2023

			0.999999999999999999920798930492018877357821248361115796849980384110811...

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

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)), A363179 phi(exp(-15*Pi)), A292864 phi(exp(-16*Pi)), A363120 phi(exp(-18*Pi)), A363021 phi(exp(-20*Pi)).

RealDigits[E^(7*Pi/12) * Gamma[1/4] * Sqrt[Sqrt[5 - Sqrt[7]] - Sqrt[3*Sqrt[7] - 7]] / (2^(13/8) * 7^(7/16) * Pi^(3/4)), 10, 120][[1]]
RealDigits[QPochhammer[E^(-14*Pi)], 10, 120][[1]]

Equals exp(7*Pi/12) * Gamma(1/4) * sqrt(sqrt(5 - sqrt(7)) - sqrt(3*sqrt(7) - 7)) / (2^(13/8) * 7^(7/16) * Pi^(3/4)).

A363120 Decimal expansion of Product_{k>=1} (1 - exp(-18Pik)).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, May 15 2023

			0.999999999999999999999999723798755647764684512427204448244366188197087...

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

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)), A363179 phi(exp(-15*Pi)), A292864 phi(exp(-16*Pi)), A363021 phi(exp(-20*Pi)).

RealDigits[QPochhammer[E^(-18*Pi)], 10, 120][[1]]
RealDigits[E^(3*Pi/4) * Gamma[1/4] * (Sqrt[6]*(2 + Sqrt[3])^(1/6) - 3)^(1/3) / (6*Pi^(3/4)), 10, 120][[1]]

Equals exp(3*Pi/4) * Gamma(1/4) * (sqrt(6)*(2 + sqrt(3))^(1/6) - 3)^(1/3) / (6*Pi^(3/4)).

A363081 Decimal expansion of Product_{k>=1} (1 - exp(-11Pik)).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, May 18 2023

			0.9999999999999990185682406467667685324890186498523246531748501440722320873182...

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

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)), A363020 phi(exp(-12*Pi)), A363178 phi(exp(-13*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)).

RealDigits[QPochhammer[E^(-11*Pi)], 10, 120][[1]]
RealDigits[QPochhammer[E^(-22*Pi)]^(5/2) / QPochhammer[E^(-44*Pi)] / EllipticTheta[3, 0, Exp[-11*Pi]]^(1/2), 10, 120][[1]]
RealDigits[E^(11*Pi/24) * Gamma[1/4] * (((-3 + Sqrt[11])^(1/4) * Sqrt[((2*(47 - 27*Sqrt[3])^(1/6) + (47 + 27*Sqrt[3])^(1/6)* Sqrt[7 + Sqrt[33]]) * (2 + ((11 + 3*Sqrt[11]) * (4 - 3*Sqrt[3] + 3*Sqrt[11]))^(1/3) + (143 + 33*Sqrt[3] + 45*Sqrt[11] + 9*Sqrt[33])^(1/3))) / (2*(47 - 27*Sqrt[3])^(1/6) * (4 + 3*Sqrt[3] + Sqrt[11]) + (47 + 27*Sqrt[3])^(1/6) * (4 - 3*Sqrt[3] + Sqrt[11]) * Sqrt[7 + Sqrt[33]])]) / (6^(7/8)*(-(11/(-4*22^(1/3) + (1490 + 837*Sqrt[3] - 351*Sqrt[11] - 306*Sqrt[33])^(1/3) + (1490 - 837*Sqrt[3] - 351*Sqrt[11] + 306*Sqrt[33])^(1/3))))^(3/8) * ((1/34012224)*(2 + (11 + 3*Sqrt[11])^(1/3) * ((4 - 3*Sqrt[3] + 3*Sqrt[11])^(1/3) + (4 + 3*Sqrt[3] + 3*Sqrt[11])^(1/3)))^12 - Sqrt[-1 + (2 + (11 + 3*Sqrt[11])^(1/3) * ((4 - 3*Sqrt[3] + 3*Sqrt[11])^(1/3) + (4 + 3*Sqrt[3] + 3*Sqrt[11])^(1/3)))^24 / 1156831381426176])^(1/8))/Pi^(3/4)), 10, 120][[1]]
RealDigits[E^(11*Pi/24) * Gamma[1/4] * Root[-387420489 + 1578379770*#1^2 - 1299078*#1^6 + 594*#1^10 + 11*#1^12 &, 2] / (Pi^(3/4) * 11^(3/8) * (2*(Root[-5832 - 3888*#1 + 1296*#1^2 + 324*#1^3 - 72*#1^4 - 12*#1^5 + #1^6 &, 2]^12 - Sqrt[-1156831381426176 + Root[-5832 - 3888*#1 + 1296*#1^2 + 324*#1^3 - 72*#1^4 - 12*#1^5 + #1^6 & , 2]^24]))^(1/8)), 10, 120][[1]]

Equals phi(exp(-22*Pi))^(5/2) / (phi(exp(-44*Pi)) * theta_3(0, exp(-11*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(11*Pi/24) * Gamma(1/4) * (((-3 + sqrt(11))^(1/4) * sqrt(((2*(47 - 27*sqrt(3))^(1/6) + (47 + 27*sqrt(3))^(1/6)* sqrt(7 + sqrt(33))) * (2 + ((11 + 3*sqrt(11)) * (4 - 3*sqrt(3) + 3*sqrt(11)))^(1/3) + (143 + 33*sqrt(3) + 45*sqrt(11) + 9*sqrt(33))^(1/3))) / (2*(47 - 27*sqrt(3))^(1/6) * (4 + 3*sqrt(3) + sqrt(11)) + (47 + 27*sqrt(3))^(1/6) * (4 - 3*sqrt(3) + sqrt(11)) * sqrt(7 + sqrt(33))))) / (6^(7/8)*(-(11/(-4*22^(1/3) + (1490 + 837*sqrt(3) - 351*sqrt(11) - 306*sqrt(33))^(1/3) + (1490 - 837*sqrt(3) - 351*sqrt(11) + 306*sqrt(33))^(1/3))))^(3/8) * ((1/34012224)*(2 + (11 + 3*sqrt(11))^(1/3) * ((4 - 3*sqrt(3) + 3*sqrt(11))^(1/3) + (4 + 3*sqrt(3) + 3*sqrt(11))^(1/3)))^12 - sqrt(-1 + (2 + (11 + 3*sqrt(11))^(1/3) * ((4 - 3*sqrt(3) + 3*sqrt(11))^(1/3) + (4 + 3*sqrt(3) + 3*sqrt(11))^(1/3)))^24 / 1156831381426176))^(1/8)) / Pi^(3/4)).

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

Original entry on oeis.org

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

Vaclav Kotesovec, May 19 2023

			0.99999999999999999816723239432842242817700113854738989073221955396667771...

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

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)).

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]]

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))).

A363179 Decimal expansion of Product_{k>=1} (1 - exp(-15Pik)).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, May 19 2023

			0.99999999999999999999657741145587875913219208544734891061914001399856284...

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

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)).

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]]

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))).

A259147 Decimal expansion of phi(exp(-Pi/2)), where phi(q) = Product_{n>=1} (1-q^n) is the Euler modular function.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 19 2015

			0.74931147780000278742962565878338031190409252790117392831206731...

Istvan Mezo, Several special values of Jacobi theta functions, arXiv:1106.2703 [math.CA], 2011-2013.
Eric Weisstein's MathWorld, Infinite Product
Eric Weisstein's MathWorld, Jacobi Theta Functions
Eric Weisstein's MathWorld, q-Pochhammer Symbol
Wikipedia, Euler function

Cf. A048651 phi(1/2), A100220 phi(1/3), A100221 phi(1/4), A100222 phi(1/5), A132034 phi(1/6), A132035 phi(1/7), A132036 phi(1/8), A132037 phi(1/9), A132038 phi(1/10), A368211 phi(exp(-Pi/16)), A292862 phi(exp(-Pi/8)), A292863 phi(exp(-Pi/4)), 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)), A259151 phi(exp(-8*Pi)), A363019 phi(exp(-10*Pi)), A363020 phi(exp(-12*Pi)), A292864 phi(exp(-16*Pi)), A363021 phi(exp(-20*Pi)).

Cf. A292819, A292823, A292827.

phi[q_] := QPochhammer[q, q]; RealDigits[phi[Exp[-Pi/2]], 10, 105] // First

phi(q) = QPochhammer(q,q) = (q;q)_infinity.
phi(q) also equals theta'(1, 0, sqrt(q))^(1/3)/(2^(1/3)*q^(1/24)), where theta' is the derivative of the elliptic theta function theta(a,u,q) w.r.t. u.
phi(exp(-Pi/2)) = ((sqrt(2) - 1)^(1/3)*(4 + 3*sqrt(2))^(1/24) * exp(Pi/48) * Gamma(1/4))/(2^(5/6)*Pi^(3/4)).
phi(exp(-Pi/2)) = (sqrt(2)-1)^(1/4) * exp(Pi/48) * Gamma(1/4)/(2^(13/16)*Pi^(3/4)). - Vaclav Kotesovec, Jul 03 2017

A000706 Expansion of modular function 1/E_3 (cf. A013973).

Original entry on oeis.org

1, 504, 270648, 144912096, 77599626552, 41553943041744, 22251789971649504, 11915647845248387520, 6380729991419236488504, 3416827666558895485479576, 1829682703808504464920468048, 979779820147442370107345764512

Offset: 0

N. J. A. Sloane

Ramanujan Lambert series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).

			G.f. = 1 + 504*q + 270648*q^2 + 144912096*q^3 + 77599626552*q^4 + 41553943041744*q^5 + ...

S. Ramanujan, Collected Papers of Srinivasa Ramanujan, pp. 115-7, Ed. G. H. Hardy et al., AMS Chelsea 2000, p. 317.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 0..360
S. Ramanujan, On the coefficients in the expansions of certain modular functions, Proc. Royal Soc., A, 95 (1918), 144-155 [G. H. Hardy, Coll. Papers, Vol. 1, 294-305.] - Added by _N. J. A. Sloane_, Feb 21 2010

Cf. A013973, A259150.

a[ n_] := SeriesCoefficient[ 1 / (1 + Sum[ -504 DivisorSigma[ 5, k] q^k, {k, n}]), {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)
a[ n_] := SeriesCoefficient[ With[ {t2 = EllipticTheta[ 2, 0, q]^4, t3 = EllipticTheta[ 3, 0, q]^4}, 1 / (t2^3 - 33 (t2 + t3) t2 t3 + t3^3)], {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)
a[ n_] := SeriesCoefficient[ With[ {t3 = EllipticTheta[ 3, 0, q]^4, t4 = EllipticTheta[ 4, 0, q]^4}, 2 / (t3^3 - 3 (t3 - t4)^2 (t3 + t4) + t4^3) ], {q, 0, 2 n}]; (* Michael Somos, Apr 26 2015 *)
a[ n_] := SeriesCoefficient[ With[ {e1 = QPochhammer[ q]^8, e4 = 32 q QPochhammer[ q^4]^8}, QPochhammer[ q^2]^12 / ((e1 + e4) (e1^2 - 16 e1 e4 - 8 e4^2)) ], {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)

{a(n) = if( n<0, 0, polcoeff( 1 / sum(k=1, n, -504*sigma(k, 5)*x^k, 1 + x * O(x^n)), n))}; /* Michael Somos, Aug 09 2007 */

{a(n) = my(A, e1, e4); if( n<0, 0, A = x * O(x^n); e1 = eta(x + A)^8; e4 = 32 * x * eta(x^4 + A)^8; polcoeff( eta(x^2 + A)^12 / ((e1 + e4) * (e1^2 - 16*e1*e4 - 8*e4^2)), n))}; /* Michael Somos, Apr 26 2015 */

Expansion of 1 / R(q) in powers of q where R() is a Ramanujan Lambert series.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^2*w^2 + 121*u^2*w^2 + 4096*u^2*v^2 - 8*v^3*w - 512*u*v^3 - 66*u*v*w^2 + 592*u*v^2*w - 4224*u^2*v*w. - Michael Somos, Aug 09 2007
Convolution inverse of A013973.
Asymptotics [Ramanujan]: a(n) ~ c * exp(2*Pi*n), where c = 2 / (96^2 * exp(-8*Pi/3) * Product_{j>=1} (1-exp(-4*Pi*j))^16) = 8192 * Pi^12 / (9 * Gamma(1/4)^16) = 0.943732053240742502013763912292610373458373085328537967959184338319972... . - Vaclav Kotesovec, Nov 08 2015

A292821 Decimal expansion of Product_{k>=1} (1 + exp(-2Pik)).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Sep 24 2017

			1.001870943123279886463534087967415218083199719502631259194986391297521...

Cf. A259149, A292825, A292829, A292819, A292820, A292822.

RealDigits[E^(Pi/12) / 2^(3/8), 10, 120][[1]]
RealDigits[QPochhammer[-1, E^(-2*Pi)]/2, 10, 120][[1]]

exp(Pi/12)/sqrtn(8,8) \\ Charles R Greathouse IV, Mar 13 2018

Equals exp(Pi/12) / 2^(3/8).

Equals A259150 / A259149.

cp's OEIS Frontend

A363117 Decimal expansion of Product_{k>=1} (1 - exp(-7*Pi*k)).

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A363118 Decimal expansion of Product_{k>=1} (1 - exp(-9*Pi*k)).

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A363119 Decimal expansion of Product_{k>=1} (1 - exp(-14*Pi*k)).

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A363120 Decimal expansion of Product_{k>=1} (1 - exp(-18*Pi*k)).

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A363081 Decimal expansion of Product_{k>=1} (1 - exp(-11*Pi*k)).

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

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A363179 Decimal expansion of Product_{k>=1} (1 - exp(-15*Pi*k)).

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A259147 Decimal expansion of phi(exp(-Pi/2)), where phi(q) = Product_{n>=1} (1-q^n) is the Euler modular function.

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A000706 Expansion of modular function 1/E_3 (cf. A013973).

A363117 Decimal expansion of Product_{k>=1} (1 - exp(-7Pik)).

A363118 Decimal expansion of Product_{k>=1} (1 - exp(-9Pik)).

A363119 Decimal expansion of Product_{k>=1} (1 - exp(-14Pik)).

A363120 Decimal expansion of Product_{k>=1} (1 - exp(-18Pik)).

A363081 Decimal expansion of Product_{k>=1} (1 - exp(-11Pik)).

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

A363179 Decimal expansion of Product_{k>=1} (1 - exp(-15Pik)).

A292821 Decimal expansion of Product_{k>=1} (1 + exp(-2Pik)).