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 1-3 of 3 results.

A034950 Expansion of eta(8z)*eta(16z)*theta_3(2z).

Original entry on oeis.org

1, 2, 0, 0, 1, -2, 0, 0, -4, -2, 0, 0, -3, 0, 0, 0, 4, -4, 0, 0, 0, 6, 0, 0, 1, 4, 0, 0, 4, 2, 0, 0, 0, -2, 0, 0, 4, -2, 0, 0, -3, 2, 0, 0, -4, -4, 0, 0, -4, 2, 0, 0, -8, -6, 0, 0, 8, -4, 0, 0, 1, -4, 0, 0, -4, 6, 0, 0, 0, 2, 0, 0, 0, -2, 0, 0, 4, 8, 0, 0, 0, 6, 0, 0, 5, -2, 0, 0, 4, -2, 0, 0, 8, 4, 0, 0, -4, -8, 0, 0, -4, 8, 0, 0, 4
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 + 2*x + x^4 - 2*x^5 - 4*x^8 - 2*x^9 - 3*x^12 + 4*x^16 - 4*x^17 + ...
G.f. = q + 2*q^3 + q^9 - 2*q^11 - 4*q^17 - 2*q^19 - 3*q^25 + 4*q^33 - ...

Links

Crossrefs

Cf. A072068, A072069, A080963.
A bisection of A248394.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 2, 0, x] EllipticTheta[ 2, Pi/4, x] / Sqrt[8 x], {x, 0, n}]; (* Michael Somos, Feb 18 2015 *)
QP = QPochhammer; s = QP[q^2]^5*(QP[q^8]/(QP[q]^2*QP[q^4])) + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^8 + A) / (eta(x + A)^2 * eta(x^4 + A)), n))}; /* Michael Somos, Feb 16 2006 */

Formula

Euler transform of period 8 sequence [2, -3, 2, -2, 2, -3, 2, -3, ...]. - Michael Somos, Feb 16 2006
Expansion of q^(-1/2) * eta(q^2)^5 * eta(q^8) / (eta(q)^2 * eta(q^4)) in powers of q. - Michael Somos, Feb 16 2006
Expansion of psi(x)^2 * psi(-x^2) = phi(x) * psi(x^2) * psi(-x^2) = phi(x) * psi(x^4) * phi(-x^4) in powers of x where phi(), psi() are Ramanujan theta functions. - Michael Somos, Feb 18 2015
G.f.: Product_{k>0} (1 + x^k)^2 * (1 - x^(2*k))^3 * (1 + x^(4*k)). - Michael Somos, Feb 16 2006
2 * a(n) = A080963(2*n + 1). a(4*n + 2) = a(4*n + 3) = 0. - Michael Somos, Feb 18 2015
a(n) = A072069(n+1) - A072068(n+1)/2. - Seichi Manymama, Sep 30 2018

A116597 Expansion of theta_3(q) * theta_4(q^4)^2 in powers of q.

Original entry on oeis.org

1, 2, 0, 0, -2, -8, 0, 0, -4, 10, 0, 0, 8, -8, 0, 0, 6, 16, 0, 0, -8, -16, 0, 0, -8, 10, 0, 0, 0, -24, 0, 0, 12, 16, 0, 0, -10, -8, 0, 0, -8, 32, 0, 0, 24, -24, 0, 0, 8, 18, 0, 0, -8, -24, 0, 0, -16, 16, 0, 0, 0, -24, 0, 0, 6, 32, 0, 0, -16, -32, 0, 0, -12, 16, 0, 0, 24, -32, 0, 0, 24, 34, 0, 0, -16, -16, 0, 0, -8, 48
Offset: 0

Views

Author

Michael Somos, Feb 18 2006

Keywords

Comments

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

Examples

			G.f. = 1 + 2*q - 2*q^4 - 8*q^5 - 4*q^8 + 10*q^9 + 8*q^12 - 8*q^13 + 6*q^16 + 16*q^17 + ...

Links

Crossrefs

Cf. A005876, A080963, A212885, A257536.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 4, 0, q^4]^2, {q, 0, n}]; (* Michael Somos, Apr 28 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * (eta(x^4 + A) / (eta(x + A) * eta(x^8 + A)))^2, n))};

Formula

Expansion of phi(q) * phi(-q^4)^2 in powers of q where phi() is a Ramanujan theta function.
Expansion of eta(q^2)^5 * (eta(q^4) / (eta(q) * eta(q^8)))^2 in powers of q.
Euler transform of period 8 sequence [ 2, -3, 2, -5, 2, -3, 2, -3, ...].
G.f.: theta_3(q) * theta_4(q^4)^2 = Product_{k>0} (1 - x^(2*k))^3 *((1 + x^k) / (1 + x^(4*k)))^2.
a(4*n + 2) = a(4*n + 3) = 0. a(n) = A080963(4*n). a(4*n) = A212885(n). a(4*n + 1) = (-1)^n * A005876(n).
a(3*n + 1) = 2 * A257536(n). - Michael Somos, Apr 28 2015

A127786 Expansion of phi(q) * phi(q^2) * phi(-q^4) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 2, 2, 4, 0, -4, 0, -8, -2, 6, -8, 4, 0, -12, 0, -8, -4, 8, 10, 12, 0, -8, 0, -8, 8, 14, -8, 16, 0, -4, 0, -16, 6, 16, 16, 8, 0, -20, 0, -8, -8, 8, -16, 20, 0, -20, 0, -16, -8, 18, 10, 8, 0, -12, 0, -24, 0, 16, -24, 12, 0, -20, 0, -24, 12, 8, 16, 28, 0, -16, 0, -8, -10, 32, -8, 20, 0, -16, 0, -16, -8, 18, 32, 20, 0, -24, 0
Offset: 0

Views

Author

Michael Somos, Jan 29 2007

Keywords

Comments

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

Examples

			G.f. = 1 + 2*q + 2*q^2 + 4*q^3 - 4*q^5 - 8*q^7 - 2*q^8 + 6*q^9 - 8*q^10 + ...

Links

Crossrefs

Cf. A033763, A080963, A116597, A212885, A213622, A246832, A246833, A246835, A246836, A246837, A246954, A257873.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2] EllipticTheta[ 4, 0, q^4], {q, 0, n}]; (* Michael Somos, Sep 08 2014 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^4 + A)^5 / (eta(x + A)^2 * eta(x^8 + A)^3), n))};

Formula

Expansion of eta(q^2)^3 * eta(q^4)^5 / (eta(q)^2 * eta(q^8)^3) in powers of q.
Euler transform of period 8 sequence [ 2, -1, 2, -6, 2, -1, 2, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 128 * (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A213622. - Michael Somos, Sep 08 2014
a(8*n + 4) = a(8*n + 6) = 0.
a(n) = A080963(2*n). a(2*n) = A116597(n). a(2*n + 1) = 2 * A246836(n). a(4*n + 1) = 2 * A246835(n). a(4*n + 3) = 4 * A246833(n). - Michael Somos, Sep 08 2014
a(8*n) = A212885(n). a(8*n + 1) = 2 * A213622(n). a(8*n + 2) = 2 * A246954(n). a(8*n + 3) = 4 * A246832(n). a(8*n + 5) = - 4 * A246837(n). a(8*n + 7) = - 8 * A033763(n). - Michael Somos, Sep 08 2014
a(3*n + 2) = 2 * A257873(n). - Michael Somos, May 11 2015
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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