A292887 -id:A292887 - cp's OEIS frontend

A292888 Decimal expansion of Product_{k>=1} (1 - exp(-3Pik)).

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

Offset: 0

Vaclav Kotesovec, Sep 26 2017

			0.999919293970017559324282655320322884698349280317277031532319284136657...

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

Cf. A292862, A292863, A259147, A259148, A259149, A259150, A259151, A292864.

Cf. A292887.

RealDigits[(5 - Sqrt[3] + Sqrt[2]*3^(3/4))^(1/6) * E^(Pi/8) * Gamma[1/4] / (2^(25/24)*3^(3/8)*Pi^(3/4)), 10, 120][[1]]
RealDigits[QPochhammer[E^(-3*Pi)], 10, 120][[1]]

(5 - sqrt(3) + sqrt(2)*3^(3/4))^(1/6) * exp(Pi/8) * gamma(1/4) / 2^(25/24) / 3^(3/8) / Pi^(3/4) \\ Charles R Greathouse IV, Sep 02 2024

Equals (5 - sqrt(3) + sqrt(2)*3^(3/4))^(1/6) * exp(Pi/8) * Gamma(1/4) / (2^(25/24) * 3^(3/8) * Pi^(3/4)).

A363018 Decimal expansion of Product_{k>=1} (1 - exp(-6Pik)).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, May 13 2023

			0.999999993487587821508587441627061243108330508136097236620870239066239...

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)), 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)), A363120 phi(exp(-18*Pi)), A363021 phi(exp(-20*Pi)).

RealDigits[E^(Pi/4)*Gamma[1/4]*(2 - Sqrt[3])^(1/12)/(2*3^(3/8)*Pi^(3/4)), 10, 120][[1]]
RealDigits[QPochhammer[E^(-6*Pi)], 10, 120][[1]]

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

Equals A292888 * A292887.

A014103 Expansion of (eta(q^2) / eta(q))^24 in powers of q.

Original entry on oeis.org

1, 24, 300, 2624, 18126, 105504, 538296, 2471424, 10400997, 40674128, 149343012, 519045888, 1718732998, 5451292992, 16633756008, 49010118656, 139877936370, 387749049720, 1046413709980, 2754808758144, 7087483527072, 17848133716832, 44056043512488, 106727749011456

Offset: 1

N. J. A. Sloane

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Given g.f. A(q), Greenhill (1895) denotes -64 * A(q^2) by tau_0 on page 409 equation (43). - Michael Somos, Jul 17 2013

			G.f. = q + 24*q^2 + 300*q^3 + 2624*q^4 + 18126*q^5 + 105504*q^6 + 538296*q^7 + ...

John H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
Albert Eagle, Elliptic functions as they should be, Galloway and Porter Ltd., Cambridge, pp. 72-73.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Kevin Acres and David Broadhurst, Eta quotients and Rademacher sums, arXiv:1810.07478 [math.NT], 2018. See Table 1 p. 10.
A. G. Greenhill, The Transformation and Division of Elliptic Functions, Proceedings of the London Mathematical Society (1895) 403-486.
R. S. Maier, On Rationally Parametrized Modular Equations, arXiv:math/0611041 [math.NT], 2006-2008, see page 4 equation (4).
MathOverflow, Identity for an infinite product [question from _T . Amdeberhan_, Oct 01 2024]
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Elliptic Lambda Function
Index entries for reversions of series

Cf. A000594, A005149, A007191, A115977.

q*mul((1+q^m)^24,m=1..30); seq(coeff(series(%,q,n+1),q,n), n=1..25);

a[ n_] := SeriesCoefficient[ q QPochhammer[ q, q^2]^-24, {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ q / Product[ 1 - q^k, {k, 1, n + 1, 2}]^24, {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := With[ {m = ModularLambda[ Log[q]/(Pi I)]}, SeriesCoefficient[ (m/16)^2 / (1 - m), {q, 0, 2 n}]]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (m/16)^2 /(1 - m), {q, 0, 2 n}]]; (* Michael Somos, Jul 11 2011 *)
eta[q_]:=q^(1/6) QPochhammer[q]; a[n_]:=SeriesCoefficient[(eta[q^2] / eta[q])^24, {q, 0, n}]; Table[a[n], {n, 4, 25}] (* Vincenzo Librandi, Oct 18 2018 *)

{a(n) = polcoeff( x * prod( k=1, n, 1 + x^k, 1 + x * O(x^n))^24, n)};

{a(n) = my(A, A2, m); if( n<0, 0, A = x + O(x^2); m=1; while( m<=n, m*=2; A = subst( A, x, x^2); A2 = A * (1 + 16*A); A = 8 * A2 + (1 + 32*A) * sqrt(A2)); polcoeff( A + 16 * A^2, n))};

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A) / eta(x + A))^24, n))};

REVERT(A005149).
Euler transform of period 2 sequence [ 24, 0, 24, 0, ... ]. - Michael Somos, Mar 19 2004
Expansion of (lambda(q) / 16)^2 / (1 - lambda(q)) in powers of q = exp(2 Pi i t) where lambda() is the elliptic modular function A115977. - Michael Somos, Nov 19 2005
Expansion of q / chi(-q)^24 in powers of q where chi() is a Ramanujan theta function.
Expansion of (theta_2(q) * theta_3(q) / (2 * theta_4(q)^2))^4 = (theta_2(q^(1/2))^2 / (4*theta_4(q^(1/2)) * theta_3(q^(1/2))))^4 in powers of q.
G.f.: x * Product_{k > 0} (1 + x^k)^24 = x / Product_{k > 0} (1 - x^(2*k - 1))^24.
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v - 48*u*v - 4096*u*v^2. - Michael Somos, Mar 19 2004
G.f. is a period 1 Fourier series which satisfies f(-1 / (2 t)) = (1/4096) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A007191. - Michael Somos, Aug 19 2007
j(q) = (f(q) + 16)^3 / f(q), j(q^2) = (f(q) + 256)^3 / f(q)^2 where j(q) is the g.f. for A000521 and f(q) is 4096 times the g.f. for a(n). - Michael Somos, Oct 01 2007
Convolution inverse of A007191. Series reversion of A005149.
Sum_{n>=1} exp(-2*Pi*n)*a(n) = 1/512. - Simon Plouffe, Feb 20 2011 [Proof: Sum_{n>=0} a(n)/exp(2*Pi*n) = exp(-2*Pi) * (phi(exp(-4*Pi)) / phi(exp(-2*Pi)))^24 = exp(-2*Pi) * A292821^24, where phi(q) is the Euler modular function. - Vaclav Kotesovec, May 13 2023]
a(n) ~ exp(2 * Pi * sqrt(2*n)) / (4096 * 2^(3/4) * n^(3/4)). - Vaclav Kotesovec, Mar 05 2015
a(1) = 1, a(n) = (24/(n-1))*Sum_{k=1..n-1} A000593(k)*a(n-k) for n > 1. - Seiichi Manyama, Apr 01 2017
G.f.: x*exp(24*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018
Expansion of Delta(q^2)/Delta(q) in powers of q where the discriminant Delta(q) is the g.f. of A000594. - Michael Somos, May 27 2022
From Vaclav Kotesovec, May 08 2023, updated May 16 2023: (Start)
Sum_{n>=1} a(n) / exp(Pi*n) = exp(-Pi) * A292820^24 = 1/8.
Sum_{n>=1} a(n) / exp(3*Pi*n) = exp(-3*Pi) * A292887^24 = (2 + sqrt(3) - sqrt(9 + 6*sqrt(3)))^8 / 512.
Sum_{n>=1} a(n) / exp(4*Pi*n) = exp(-4*Pi) * A292822^24 = 99*sqrt(2)/2048 - 35/512.
Sum_{n>=1} a(n) / exp(5*Pi*n) = exp(-5*Pi) * A292904^24 = (2 + sqrt(5) - sqrt((15 + 7*sqrt(5))/2))^12 / 8.
Sum_{n>=1} a(n) / exp(6*Pi*n) = exp(-6*Pi) * (A363020/A363018)^24 = 385/512 + 7*sqrt(3)/16 - sqrt(74664 + 43134*sqrt(3))/256.
Sum_{n>=1} a(n) / exp(7*Pi*n) = exp(-7*Pi) * (A363119/A363117)^24 = (sqrt(7) - 1 - sqrt(22*sqrt(7) - 56))^3 / (2^(15/2) * (2^(1/4)*sqrt(5 + sqrt(7)) + (56 + 23*sqrt(7))^(1/4))^6).
Sum_{n>=1} a(n) / exp(8*Pi*n) = exp(-8*Pi) * (A292864/A259151)^24 = -8963/512 - 99*sqrt(2)/8 + 9*sqrt(126913704 + 89741542*sqrt(2))/4096.
Sum_{n>=1} a(n) / exp(9*Pi*n) = exp(-9*Pi) * (A363120/A363118)^24 = ((6*(3 + sqrt(3)))^(1/3) - 3)^8 / (8*((3*(6 + 7*sqrt(3) + 3*sqrt(14*sqrt(3) - 15)))^(1/3) - 3)^8).
Sum_{n>=1} a(n) / exp(10*Pi*n) = exp(-10*Pi) * (A363021/A363019)^24 = (5^(1/4) - 1)^24 / 2097152.
Sum_{n>=1} a(n) / exp(Pi*n/2) = exp(-Pi/2) * A292819^24 = 35 + 99/2^(3/2).
Sum_{n>=1} (-1)^(n+1) * a(n) / exp(Pi*n) = 1/64.
Sum_{n>=1} (-1)^(n+1) * a(n) / exp(2*Pi*n) = 99/2^(3/2) - 35.
Sum_{n>=1} (-1)^(n+1) * a(n) / exp(3*Pi*n) = 97/64 - 7*sqrt(3)/8.
Sum_{n>=1} (-1)^(n+1) * a(n) / exp(4*Pi*n) = -5018696 - 3548754*sqrt(2) + (9/2)*sqrt(2487635528172 + 1759023951091*sqrt(2)).
Sum_{n>=1} (-1)^(n+1) * a(n) / exp(7*Pi*n) = 13880161/64 + 81972*sqrt(7) - 9*sqrt(74328271227 + 28093445864*sqrt(7))/8. (End)
The g.f. A(q) satisfies -(16)^2 * A(q^2) = (lambda(q) + lambda(-q)) = (lambda(q)*lambda(-q)), where lambda(q) = 16*q - 128*q^2 + 704*q^3 - ... is the elliptic modular function in powers of the nome q = exp(i*Pi*t), the g.f. of A115977; lambda(q) = k(q)^2, where k(q) = (theta_2(q) / theta_3(q))^2 is the elliptic modulus. - Peter Bala, Sep 26 2023
From Peter Bala, Sep 26 2023: (Start)
A(q^2) = -A(q)*A(-q).
A(q) = lambda(-q)^2/(16*lambda(q)) = lambda(-q)*(lambda(-q) -  1)/16. (End)
G.f. A(x) satisfies 0 = f(A(x), A(-x)) where f(u, v) = u + v + 48*u*v - 4096*u^2*v^2. - Michael Somos, Oct 07 2024

More terms from Michael Somos, Nov 24 2001

A292904 Decimal expansion of Product_{k>=1} (1 + exp(-5Pik)).

Original entry on oeis.org

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, 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, 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

Offset: 1

Vaclav Kotesovec, Sep 26 2017

			1.000000150701750250023989493869871467973761006430507405690199988520887...

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
Index entries for transcendental numbers

Cf. A292819, A292820, A292821, A292887, A292822, A292905.

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]]
RealDigits[QPochhammer[-1, E^(-5*Pi)]/2, 10, 120][[1]]
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 *)

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

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.

Equals exp(5*Pi/24) * sqrt(2 + sqrt(5) - sqrt((15 + 7*sqrt(5))/2))/2^(1/8). - Vaclav Kotesovec, May 13 2023

cp's OEIS Frontend

A292888 Decimal expansion of Product_{k>=1} (1 - exp(-3*Pi*k)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A363018 Decimal expansion of Product_{k>=1} (1 - exp(-6*Pi*k)).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A014103 Expansion of (eta(q^2) / eta(q))^24 in powers of q.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

Formula

Extensions

A292904 Decimal expansion of Product_{k>=1} (1 + exp(-5*Pi*k)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A292888 Decimal expansion of Product_{k>=1} (1 - exp(-3Pik)).

A363018 Decimal expansion of Product_{k>=1} (1 - exp(-6Pik)).

A292904 Decimal expansion of Product_{k>=1} (1 + exp(-5Pik)).