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.

Showing 1-4 of 4 results.

A001938 Expansion of k/(4*q^(1/2)) in powers of q, where k defined by sqrt(k) = theta_2(0, q)/theta_3(0, q).

Original entry on oeis.org

1, -4, 14, -40, 101, -236, 518, -1080, 2162, -4180, 7840, -14328, 25591, -44776, 76918, -129952, 216240, -354864, 574958, -920600, 1457946, -2285452, 3548550, -5460592, 8332425, -12614088, 18953310, -28276968, 41904208, -61702876, 90304598, -131399624
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
k^2 is the parameter and q the Jacobi nome of elliptic functions. See, e.g., Fricke, p. 11, eq. (8) with p. 10. eq. (1). - Wolfdieter Lang, Jul 04 2016

Examples

			G.f. = 1 - 4*x + 14*x^2 - 40*x^3 + 101*x^4 - 236*x^5 + 518*x^6 - 1080*x^7 + ...
G.f. of B(q) = q * A(q^2): q - 4*q^3 + 14*q^5 - 40*q^7 + 101*q^9 - 236*q^11 + 518*q^13 - 1080*q^15 + ...

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.
  • E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable, 1935, Oxford University Press, p. 385.
  • N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).
  • 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).

Links

Crossrefs

Cf. A000122, A000700, A001936, A010054, A079006, A093160, A121373, A127393, A127931, A127932, A139820.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ -x, x^2] QPochhammer[ x^2, x^4])^4, {x, 0, n}]; (* Michael Somos, Sep 24 2011 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^4] / QPochhammer[ -x])^4, {x, 0, n}]; (* Michael Somos, Sep 24 2011 *)
a[ n_] := SeriesCoefficient[ (Product[ 1 - x^k, {k, 4, n, 4}] / Product[ 1 - (-x)^k, {k, n}])^4, {x, 0, n}]; (* Michael Somos, Sep 24 2011 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q^(1/2)] / (2 EllipticTheta[ 3, 0, q]))^4, {q, 0, n + 1/2}]; (* Michael Somos, Sep 24 2011 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q] / EllipticTheta[ 2, 0, q^(1/2)])^4, {q, 0, n + 1/2}]; (* Michael Somos, Sep 24 2011 *)
  • PARI
    {a(n) = my(A, A2, m); if( n<0, 0, n = 2*n + 1; A = x + O(x^3); m=2; while( mMichael Somos, Mar 26 2004 */
  • 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)^3)^4, n))}; /* Michael Somos, Mar 26 2004 */

Formula

Expansion of (psi(x^2) / phi(x))^2 = (psi(x) / phi(x))^4 = (psi(x^2) / psi(x))^4 = (psi(-x) / psi(-x^2))^4 = (chi(-x) / chi(-x^2)^2)^4 = (chi(x)^2 * chi(-x))^-4 = (chi(x) * chi(-x^2))^-4 = (f(-x^4) / f(x))^4 in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Feb 26 2012
G.f. A(x) satisfies 1 = (1 - 16 * x * A(x)^2) * (1 + 16 * x * A(-x)^2). - Michael Somos, Mar 26 2004
Expansion of q^(-1/2) * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^4 in powers of q.
Euler transform of period 4 sequence [ -4, 8, -4, 0, ...].
Given g.f. A(x), then B(q) = q * A(q^2) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = v - (u * (1 + 4*v))^2. - Michael Somos, Mar 26 2004
G.f. A(q) satisfies A(q) = sqrt(A(q^2)) / (1 + 4*q*A(q^2)); together with limit_{n->infinity} A(x^n) = 1 this gives a fast algorithm to compute the series. - Joerg Arndt, Aug 06 2011
G.f.: (Product_{k>0} (1 + x^(2*k)) / (1 + x^(2*k - 1)))^4.
a(n) = (-1)^n * A093160(n). Convolution square of A079006.
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 1/4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A139820. - Michael Somos, Jun 04 2015
G.f.: ((Sum_{n >= 0} x^(n*(n+1))) / (1 + Sum_{n >= 1} x^(n^2)))^4 (from the sum representation of the Jacobi theta functions evaluated at vanishing argument). - Wolfdieter Lang, Jul 04 2016
a(n) ~ (-1)^n * exp(sqrt(2*n)*Pi) / (32 * 2^(1/4) * n^(3/4)). - Vaclav Kotesovec, Nov 15 2017

Extensions

Edited by N. J. A. Sloane, Mar 31 2007

A014969 Expansion of (theta_3(q) / theta_4(q))^2 in powers of q.

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
Offset: 0

Views

Author

N. J. A. Sloane

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

  • A. Cayley, An Elementary Treatise on Elliptic Functions, 2nd ed, 1895, p. 380, Section 488.
  • J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
  • N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).
  • R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see p. 375. Eqs. (17),(18),(19).

Links

Crossrefs

Cf. A014972, A029841, A107035, A115977, A131126, A139820.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ 1 / Sqrt[1 - InverseEllipticNomeQ  @ q], {q, 0, n}]; (* Michael Somos, Aug 01 2011 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] / EllipticTheta[ 4, 0, q])^2, {q, 0, n}]; (* Michael Somos, Aug 01 2011 *)
nmax=60; CoefficientList[Series[Product[((1+x^(2*k+1))/(1-x^(2*k+1)))^4,{k,0,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Aug 28 2015 *)
s = (QPochhammer[q^2]^3/(QPochhammer[q]^2*QPochhammer[q^4]))^4+O[q]^30; CoefficientList[s, q] (* Jean-François Alcover, Nov 09 2015, adapted from PARI *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 / (eta(x + A)^2 * eta(x^4 + A)))^4, n))}; /* Michael Somos, Jul 07 2005 */

Formula

Expansion of (phi(q) / phi(-q))^2 = (phi(q) / phi(-q^2))^4 = (phi(-q^2) / phi(-q))^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() are Ramanujan theta functions. - Michael Somos, Aug 01 2011
Expansion of Fricke t(omega) = tau(omega) / 2 + 1 in powers of q = exp(2 Pi i omega).
Expansion of elliptic 1 / sqrt(1 - lambda(q)) = 1 / k'(q) in powers of the nome q = exp(Pi*i*z).
Euler transform of period 4 sequence [ 8, -4, 8, 0, ...]. - Michael Somos, Jul 07 2005
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (1 + u)^2 - 4*u*v^2. - Michael Somos, Nov 14 2006
G.f.: (theta_3(x) / theta_4(x))^2 = (Sum_{k} x^k^2) / (Sum_{k} (-x)^k^2)^2 = (Product_{k>0} (1 - x^(4*k - 2)) / ((1 - x^(4*k - 1)) * (1 - x^(4*k - 3)))^2)^4.
A139820(n) = (-1)^n * a(n). 8 * A107035(n) = a(n) unless n=0. 2 * A131126(n) = a(n) unless n=0. Convolution inverse of A139820.
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 1/4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A029841. - Michael Somos, Jun 04 2015
a(n) ~ exp(Pi*sqrt(2*n)) / (8 * 2^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 28 2015
G.f.: exp(8*Sum_{k>=1} sigma(2*k - 1)*x^(2*k-1)/(2*k - 1)). - Ilya Gutkovskiy, Apr 19 2019
Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = (1/4)*sqrt(8 + 6*sqrt(2)). - Simon Plouffe, Mar 02 2021
From Peter Bala, Sep 25 2023: (Start)
G.f.: A(q) = sqrt(-lambda(-q)/lambda(q)), where lambda(q) = 16*q - 128*q^2 + 704*q^3 - 3072*q^4 + ... 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.
A(q) = sqrt(G(q)), where G(q) = 1 + 16q + 128*q^2 + 704*q^3 + 3072*q^4 + ... is the g.f. of A014972. (End)

A097243 Expansion of 1 + 32 * (eta(q^4) / eta(q))^8 in powers of q.

Original entry on oeis.org

1, 32, 256, 1408, 6144, 22976, 76800, 235264, 671744, 1809568, 4640256, 11404416, 27009024, 61905088, 137803776, 298806528, 632684544, 1310891584, 2662655232, 5310231424, 10412576768, 20098970624, 38231811072, 71734039808, 132875747328, 243175399136
Offset: 0

Views

Author

Michael Somos, Aug 02 2004

Keywords

Comments

Expansion of a q-series used in construction of j(tau) to j(2tau) iteration.

Examples

			G.f. = 1 + 32*x + 256*x^2 + 1408*x^3 + 6144*x^4 + 22976*x^5 + 76800*x^6 + ...

References

  • H. Cohn, Introduction to the construction of class fields, Cambridge 1985, p. 191

Links

Crossrefs

Cf. A007248, A007096, A014969, A092877, A139820, A189925, A212318, A232358.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ 1 + 32 x (QPochhammer[ x^4] / QPochhammer[ x])^8, {x, 0, n}]; (* Michael Somos, Dec 15 2016 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x^n * O(x); polcoeff( 1 + 32 * x * (eta(x^4 + A) / eta(x + A))^8, n))};

Formula

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u+3)^2 - 8*(u+1)*v^2.
a(n) = 32*A092877(n), if n>0. a(n) = A007096(4*n).
a(n) = A014969(2*n) = A139820(2*n) = A189925(4*n) = A212318(4*n) = A232358(4*n). - Michael Somos, Dec 15 2016
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 1/8 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A007248. - Michael Somos, Dec 15 2016
a(n) ~ exp(2*Pi*sqrt(n))/(16*n^(3/4)). - Vaclav Kotesovec, Sep 08 2017

A260145 Expansion of x * (psi(x^4) / phi(x))^2 in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, -4, 12, -32, 78, -176, 376, -768, 1509, -2872, 5316, -9600, 16966, -29408, 50088, -83968, 138738, -226196, 364284, -580032, 913824, -1425552, 2203368, -3376128, 5130999, -7738136, 11585208, -17225472, 25444278, -37350816, 54504160, -79085568, 114133296
Offset: 1

Views

Author

Michael Somos, Jul 17 2015

Keywords

Comments

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

Examples

			G.f. = x - 4*x^2 + 12*x^3 - 32*x^4 + 78*x^5 - 176*x^6 + 376*x^7 + ...

Links

Crossrefs

Cf. A014103, A022577, A092877, A107035, A131125, A139820, A210063, A210066, A210067.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (1/4) EllipticTheta[ 2, 0, q^2]^2 / EllipticTheta[ 3, 0, q]^2, {q, 0, n}];
  • PARI
    {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A)^2 * eta(x^4 + A) * eta(x^8 + A)^2 / eta(x^2 + A)^5)^2, n))};

Formula

Expansion of (eta(q)^2 * eta(q^4) * eta(q^8)^2 / eta(q^2)^5)^2 in powers of q.
Euler transform of period 8 sequence [ -4, 6, -4, 4, -4, 6, -4, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 1/8 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A210067.
G.f.: x * Product_{k>0} ( 1 + x^(2*k))^6 * (1 + x^(4*k))^4 / (1 + x^k)^4.
a(n) = -(-1)^n * A107035(n). -4 * a(n) = A210066(n) unless n=0. -8 * a(n) = A139820(n) unless n=0.
a(2*n) = -4 * A092877(n). a(2*n + 1) = A022577(n). a(4*n) = -32 * A014103(n).
Convolution square of A210063. Convolution inverse of A131125.
a(n) ~ -(-1)^n * exp(sqrt(2*n)*Pi) / (64 * 2^(1/4) * n^(3/4)). - Vaclav Kotesovec, Nov 17 2017
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