cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A127392 Expansion of the elliptic function sqrt(k(q))/q^(1/4) in powers of q, where sqrt(k(q)) = theta_2(q)/theta_3(q).

Original entry on oeis.org

2, -4, 10, -20, 36, -64, 110, -180, 288, -452, 692, -1044, 1554, -2276, 3296, -4724, 6696, -9408, 13108, -18112, 24850, -33864, 45844, -61696, 82564, -109892, 145536, -191828, 251684, -328804, 427802, -554408, 715808, -920896, 1180660, -1508736, 1921896, -2440740, 3090612
Offset: 0

Views

Author

N. J. A. Sloane, Mar 31 2007

Keywords

Examples

			2 - 4*x + 10*x^2 - 20*x^3 + 36*x^4 - 64*x^5 + 110*x^6 - 180*x^7 + 288*x^8 - ...
2*q^(1/4) - 4*q^(5/4) + 10*q^(9/4) - 20*q^(13/4) + 36*q^(17/4) - 64*q^(21/4) + ...

References

  • N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).

Crossrefs

See A127391 for another version. Dividing by 2 gives A079006. Cf. A001936, A001938.

Programs

  • Mathematica
    QP = QPochhammer; s = 2*(QP[q]*(QP[q^4]^2/QP[q^2]^3))^2 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015, adapted from PARI *)
nmax = 50; CoefficientList[Series[2*Product[(1+x^(2*k))^4 / (1+x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 04 2016 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( 2 * (eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^3)^2, n))} /* Michael Somos, Jun 12 2012 */

Formula

Expansion of 2 * q^(-1/4) * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^2 in powers of q. - Michael Somos, Jun 12 2012
a(n) ~ (-1)^n * exp(Pi*sqrt(n))/(2^(5/2)*n^(3/4)). - Vaclav Kotesovec, Jul 04 2016
2 * ({1} U (Euler transform of period 4 sequence [-2, 4, -2, 0])). - Georg Fischer, Dec 06 2022

A139820 Expansion of (phi(-q) / phi(q))^2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -8, 32, -96, 256, -624, 1408, -3008, 6144, -12072, 22976, -42528, 76800, -135728, 235264, -400704, 671744, -1109904, 1809568, -2914272, 4640256, -7310592, 11404416, -17626944, 27009024, -41047992, 61905088, -92681664, 137803776, -203554224
Offset: 0

Views

Author

Michael Somos, May 01 2008

Keywords

Comments

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

Examples

			G.f. = 1 - 8*q + 32*q^2 - 96*q^3 + 256*q^4 - 624*q^5 + 1408*q^6 - 3008*q^7 + ...

References

  • N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).

Links

Crossrefs

Cf. A001938, A014969.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ Sqrt[1 - InverseEllipticNomeQ [ q]], {q, 0, n}];
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q] / EllipticTheta[ 3, 0, q])^2, {q, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^2 * eta(x^4 + A) / eta(x^2 + A)^3)^4, n))};

Formula

Expansion of (eta(q)^2 * eta(q^4) / eta(q^2)^3)^4 in powers of q.
Expansion of (phi(-q) / phi(q))^2 = (phi(-q^2) / phi(q))^4 = (phi(-q) / phi(-q^2))^4 = (psi(-q) / psi(q))^4 = (chi(-q^2) / chi(q)^2)^4 = (chi(-q) / chi(q))^4 = (chi(-q)^2 / chi(-q^2))^4 = (f(-q) / f(q))^4 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
Expansion of Jacobian elliptic function k'(q) in powers of nome q.
Euler transform of period 4 sequence [ -8, 4, -8, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = 4 * u - v^2 * (1 + u)^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 4 g(t) where q = exp(2 Pi i t) and g() is g.f. for A001938.
G.f.: ((Sum_{k} (-x)^k^2) / (Sum_{k} x^k^2))^2 = (Product_{k>0} (1 + x^(2*k)) / (1 + x^k)^2)^4.
a(n) = (-1)^n * A014969(n). Convolution inverse of A014969.
Empirical : Sum_{n>=1} exp(-Pi)^(n-1)*(-1)^(n+1)*a(n) = sqrt(2). - Simon Plouffe, Feb 20 2011
a(n) ~ (-1)^n * exp(Pi*sqrt(2*n)) / (8 * 2^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 08 2017
G.f.: exp(-8*Sum_{k>=1} sigma(2*k - 1)*x^(2*k-1)/(2*k - 1)). - Ilya Gutkovskiy, Apr 19 2019

A195861 Expansion of (psi(x) / phi(x))^5 in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, -5, 20, -65, 185, -481, 1165, -2665, 5820, -12220, 24802, -48880, 93865, -176125, 323685, -583798, 1035060, -1806600, 3108085, -5276305, 8846884, -14663645, 24044285, -39029560, 62755345, -100004806, 158022900, -247710570, 385366265
Offset: 0

Views

Author

Michael Somos, Sep 24 2011

Keywords

Comments

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

Examples

			G.f. = 1 - 5*x + 20*x^2 - 65*x^3 + 185*x^4 - 481*x^5 + 1165*x^6 - 2665*x^7 + ...
G.f. = q^5 - 5*q^13 + 20*q^21 - 65*q^29 + 185*q^37 - 481*q^45 + 1165*q^53 - 2665*q^61 + ...

References

  • A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.

Links

Crossrefs

Cf. A001939, A029842.
(psi(x) / phi(x))^b: A083365 (b=1), A079006 (b=2), A187053 (b=3), A001938 (b=4), this sequence (b=5).

Programs

  • Mathematica
    a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (m / 16)^(5/8), {q, 0, n + 5/8}]];
a[ n_] := SeriesCoefficient[ Product[(1 + x^(k + 1)) / (1 + x^k), {k, 1, n, 2}]^5, {x, 0, n}];
a[ n_] := SeriesCoefficient[ (QPochhammer[ -x^2, x^2] / QPochhammer[ -x, x^2])^5, {x, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^2)^5, n))};

Formula

Expansion of q^(-5/8) * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^5 in powers of q.
Euler transform of period 4 sequence [-5, 10, -5, 0, ...].
G.f.: (Product_{k>0} (1 + x^(2*k)) / (1 + x^(2*k - 1)))^5.
a(n) = (-1)^n * A001939(n). Convolution inverse of A029842.
a(n) ~ (-1)^n * 5^(1/4) * exp(sqrt(5*n/2)*Pi) / (64 * 2^(3/4) * n^(3/4)). - Vaclav Kotesovec, Nov 27 2015

A134746 Expansion of 1+k in powers of q^(1/2) where q is Jacobi's nome and k is the elliptic modulus.

Original entry on oeis.org

1, 4, 0, -16, 0, 56, 0, -160, 0, 404, 0, -944, 0, 2072, 0, -4320, 0, 8648, 0, -16720, 0, 31360, 0, -57312, 0, 102364, 0, -179104, 0, 307672, 0, -519808, 0, 864960, 0, -1419456, 0, 2299832, 0, -3682400, 0, 5831784, 0, -9141808, 0, 14194200, 0, -21842368, 0
Offset: 0

Views

Author

Michael Somos, Nov 07 2007

Keywords

Comments

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

Examples

			G.f. = 1 + 4*q - 16*q^3 + 56*q^5 - 160*q^7 + 404*q^9 - 944*q^11 + 2072*q^13 + ...

Links

Crossrefs

Cf. A001938, A127393, A210066, A210067.

Programs

  • Mathematica
    CoefficientList[Series[(QPochhammer[x^8]/QPochhammer[x])^4 (QPochhammer[x^2]/QPochhammer[x^4])^14, {x, 0, 50}], x] (* Jan Mangaldan, Mar 21 2013 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] / EllipticTheta[ 3, 0, q^2])^2, {q, 0, n}]; (* Michael Somos, Oct 16 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( (eta(x^8 + A) / eta(x + A))^2 * (eta(x^2 + A) / eta(x^4 + A))^7 )^2, n))};

Formula

Expansion of (phi(q) / phi(q^2))^2 in powers of q where phi() is a Ramanujan theta function.
Expansion of (eta(q^8) / eta(q))^4 * (eta(q^2) / eta(q^4))^14 in powers of q.
Euler transform of period 8 sequence [ 4, -10, 4, 4, 4, -10, 4, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A210066.
G.f.: ( (Sum_{k in Z} x^(k^2)) / (Sum_{k in Z} x^(2*k^2)) )^2 = ( Product_{k>0} (1 + x^k)^2 * (1 + x^(4*k))^2 / (1 + x^(2*k))^5 )^2.
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (2 - v)^2 - u * (2 - u) * v^2.
a(2*n) = 0 unless n=0. a(2*n + 1) = 4 * A001938(n) = A127393(n).
a(n) = (-1)^n * A210067(n). Convolution inverse of A210066. - Michael Somos, Oct 16 2015
Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = 34 + 24*sqrt(2) - 4*sqrt(140 + 99*sqrt(2)). - Simon Plouffe, Mar 04 2021

A127393 Expansion of k/q^(1/2) in powers of q, where k is the elliptic function defined by sqrt(k) = theta_2/theta_3.

Original entry on oeis.org

4, -16, 56, -160, 404, -944, 2072, -4320, 8648, -16720, 31360, -57312, 102364, -179104, 307672, -519808, 864960, -1419456, 2299832, -3682400, 5831784, -9141808, 14194200, -21842368, 33329700, -50456352, 75813240, -113107872, 167616832, -246811504, 361218392, -525598496
Offset: 0

Views

Author

N. J. A. Sloane, Apr 01 2007

Keywords

Comments

The elliptic modulus k is often used in elliptic integrals. - Michael Somos, Jun 11 2017

Examples

			G.f. = 4 - 16*x + 56*x^2 - 160*x^3 + 404*x^4 - 944*x^5 + ... - _Michael Somos_, Jan 26 2025

Crossrefs

See A001938, the main entry for this sequence, for further information.

Formula

a(n) = 4*A001938(n).
k = 4*q^(1/2) - 16*q^(3/2) + 56*q^(5/2) - 160*q^(7/2) + ... where the nome q = e^(-Pi*K'/K). - Michael Somos, Jun 11 2017

A210067 Expansion of (phi(-q) / phi(q^2))^2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -4, 0, 16, 0, -56, 0, 160, 0, -404, 0, 944, 0, -2072, 0, 4320, 0, -8648, 0, 16720, 0, -31360, 0, 57312, 0, -102364, 0, 179104, 0, -307672, 0, 519808, 0, -864960, 0, 1419456, 0, -2299832, 0, 3682400, 0, -5831784, 0, 9141808, 0, -14194200, 0, 21842368, 0
Offset: 0

Views

Author

Michael Somos, Mar 16 2012

Keywords

Comments

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

Examples

			1 - 4*q + 16*q^3 - 56*q^5 + 160*q^7 - 404*q^9 + 944*q^11 - 2072*q^13 + ...

Links

Crossrefs

Cf. A001938, A127393, A131126, A134746, A210030.

Programs

  • Mathematica
    a[n_] := SeriesCoefficient[(EllipticTheta[3, 0, -q]/EllipticTheta[3, 0, q^2])^2, {q, 0, n}]; Table[a[n], {n,0,50}] (* G. C. Greubel, Nov 29 2017 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^2 * eta(x^2 + A) * eta(x^8 + A)^2 / eta(x^4 + A)^5)^2, n))}

Formula

Expansion of (eta(q)^2 * eta(q^2) * eta(q^8)^2 / eta(q^4)^5)^2 in powers of q.
Euler transform of period 8 sequence [ -4, -6, -4, 4, -4, -6, -4, 0, ...].
a(2*n) = 0 unless n=0. a(2*n + 1) = -4 * A001938(n) = -A127393(n).
a(n) = (-1)^n * A134746(n).
Convolution inverse of A131126. Convolution square of A210030.
Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = -32 - 24*sqrt(2) + 4*sqrt(140+99*sqrt(2)). - Simon Plouffe, Mar 02 2021

A320049 Expansion of (psi(x) / phi(x))^6 in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, -6, 27, -98, 309, -882, 2330, -5784, 13644, -30826, 67107, -141444, 289746, -578646, 1129527, -2159774, 4052721, -7474806, 13569463, -24274716, 42838245, -74644794, 128533884, -218881098, 368859591, -615513678, 1017596115, -1667593666, 2710062756, -4369417452
Offset: 0

Views

Author

Seiichi Manyama, Oct 04 2018

Keywords

Links

Crossrefs

(psi(x) / phi(x))^b: A083365 (b=1), A079006 (b=2), A187053 (b=3), A001938 (b=4), A195861 (b=5), this sequence (b=6), A320050 (b=7).
Cf. A029843.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[Product[((1-x^k) * (1-x^(4*k))^2 / (1-x^(2*k))^3)^6, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 06 2018 *)

Formula

Convolution inverse of A029843.
Expansion of q^(-3/4) * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^6 in powers of q.
a(n) ~ (-1)^n * 3^(1/4) * exp(Pi*sqrt(3*n)) / (128*sqrt(2)*n^(3/4)). - Vaclav Kotesovec, Oct 06 2018

A320050 Expansion of (psi(x) / phi(x))^7 in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, -7, 35, -140, 483, -1498, 4277, -11425, 28889, -69734, 161735, -362271, 786877, -1662927, 3428770, -6913760, 13660346, -26492361, 50504755, -94766875, 175221109, -319564227, 575387295, -1023624280, 1800577849, -3133695747, 5399228149, -9214458260, 15584195428
Offset: 0

Views

Author

Seiichi Manyama, Oct 04 2018

Keywords

Comments

In general, for b > 0 and (psi(x) / phi(x))^b, a(n) ~ (-1)^n * b^(1/4) * exp(Pi*sqrt(b*(n/2))) / (2^(b + 7/4) * n^(3/4)). - Vaclav Kotesovec, Oct 06 2018

Links

Crossrefs

(psi(x) / phi(x))^b: A083365 (b=1), A079006 (b=2), A187053 (b=3), A001938 (b=4), A195861 (b=5), A320049 (b=6), this sequence (b=7).
Cf. A029844.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[((1-x^k) * (1-x^(4*k))^2 / (1-x^(2*k))^3)^7, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 06 2018 *)

Formula

Convolution inverse of A029844.
Expansion of q^(-7/8) * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^7 in powers of q.
a(n) ~ (-1)^n * 7^(1/4) * exp(Pi*sqrt((7*n)/2)) / (256*2^(3/4)*n^(3/4)). - Vaclav Kotesovec, Oct 06 2018

A216060 Expansion of (phi(q) / phi(q^4))^2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 4, 4, 0, 0, -8, -16, 0, 0, 20, 56, 0, 0, -40, -160, 0, 0, 72, 404, 0, 0, -128, -944, 0, 0, 220, 2072, 0, 0, -360, -4320, 0, 0, 576, 8648, 0, 0, -904, -16720, 0, 0, 1384, 31360, 0, 0, -2088, -57312, 0, 0, 3108, 102364, 0, 0, -4552, -179104, 0, 0, 6592
Offset: 0

Views

Author

Michael Somos, Aug 31 2012

Keywords

Comments

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

Examples

			1 + 4*q + 4*q^2 - 8*q^5 - 16*q^6 + 20*q^9 + 56*q^10 - 40*q^13 - 160*q^14 + ...

Links

Crossrefs

Cf. A001938, A079006, A208274.

Programs

  • Mathematica
    a[n_]:= SeriesCoefficient[(EllipticTheta[3, 0, q]/EllipticTheta[3, 0, q^4])^2, {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Dec 04 2017 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^5 * eta(x^16 + A)^2 / (eta(x + A)^2 * eta(x^8 + A)^5))^2, n))}

Formula

Expansion of (eta(q^2)^5 * eta(q^16)^2 / (eta(q)^2 * eta(q^8)^5))^2 in powers of q.
Euler transform of period 16 sequence [ 4, -6, 4, -6, 4, -6, 4, 4, 4, -6, 4, -6, 4, -6, 4, 0, ...].
a(4*n) = 0 unless n=0. a(4*n + 3) = 0. a(4*n + 1) = 4 * A079006(n). a(4*n + 2) = 4 * A001938(n).
Convolution square of A208274.
Empirical: Sum{n>=0} a(n)/exp(Pi*n) = 40 + 28*sqrt(2) - 8*sqrt(48+34*sqrt(2)). - Simon Plouffe, Mar 02 2021

A225915 Expansion of (k(q) / 4)^4 in powers of q where k() is a Jacobi elliptic function.

Original entry on oeis.org

1, -16, 152, -1088, 6444, -33184, 153152, -646528, 2533070, -9311664, 32387616, -107299904, 340436664, -1039026144, 3061896704, -8739810688, 24229115109, -65390485328, 172155210320, -442928464640, 1115433685796, -2753362613984, 6670224790272, -15876957230848
Offset: 2

Views

Author

Michael Somos, May 20 2013

Keywords

Examples

			G.f. = q^2 - 16*q^3 + 152*q^4 - 1088*q^5 + 6444*q^6 - 33184*q^7 + 153152*q^8 + ...

Links

Crossrefs

Cf. A001938, A005798, A079006, A083365.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (InverseEllipticNomeQ[  q] / 16)^2, {q, 0, n}];
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q] / EllipticTheta[ 2, 0, q^(1/2)])^16, {q, 0, n}];
a[ n_] := SeriesCoefficient[ q ( Product[ 1 - q^k, {k, 4, n - 1, 4}]/
Product[ 1 - (-q)^k, {k, n - 1}])^16, {q, 0, n}];
  • PARI
    {a(n) = my(A); if( n<2, 0, n-=2; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^3)^16, n))};

Formula

Expansion of (eta(q) * eta(q^4)^2 / eta(q^2)^3)^16 in powers of q.
Euler transform of period 4 sequence [-16, 32, -16, 0, ...].
G.f.: q^2 * (Product_{k>0} (1 + q^(2*k)) / (1 + q^(2*k - 1)))^16.
Convolution square of A005798.
a(n) ~ (-1)^n * exp(2*Pi*sqrt(2*n)) / (65536 * 2^(3/4) * n^(3/4)). - Vaclav Kotesovec, Nov 17 2017
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