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 1-5 of 5 results.

A213625 Expansion of psi(x)^2 * phi(x^2) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 2, 3, 6, 4, 4, 7, 2, 8, 10, 4, 10, 9, 6, 8, 10, 4, 8, 16, 8, 9, 12, 8, 12, 20, 6, 8, 10, 8, 18, 11, 12, 8, 20, 12, 8, 20, 6, 20, 26, 8, 8, 15, 10, 16, 18, 12, 16, 20, 10, 16, 16, 8, 24, 24, 8, 21, 26, 8, 20, 20, 14, 8, 28, 16, 10, 28, 10, 24, 22, 8, 16, 17
Offset: 0

Views

Author

Michael Somos, Jun 16 2012

Keywords

Comments

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

Examples

			G.f. = 1 + 2*x + 3*x^2 + 6*x^3 + 4*x^4 + 4*x^5 + 7*x^6 + 2*x^7 + 8*x^8 + 10*x^9 + ...
G.f. = q + 2*q^5 + 3*q^9 + 6*q^13 + 4*q^17 + 4*q^21 + 7*q^25 + 2*q^29 + 8*q^33 + ...

Links

Crossrefs

Cf. A116597, A132969, A213622.

Programs

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

Formula

Expansion of q^(-1/4) * eta(q^2)^2 * eta(q^4)^5 / (eta(q)^2 * eta(q^8)^2) in powers of q.
Euler transform of period 8 sequence [ 2, 0, 2, -5, 2, 0, 2, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 2^(3/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A116597.
a(2*n) = A213622(n). a(2*n + 1) = 2 * A132969(n).

A033717 Number of integer solutions to the equation x^2 + 2*y^2 + 4*z^2 = n.

Original entry on oeis.org

1, 2, 2, 4, 4, 4, 8, 8, 6, 6, 8, 4, 8, 12, 0, 8, 12, 8, 10, 12, 8, 8, 24, 8, 8, 14, 8, 16, 16, 4, 0, 16, 6, 16, 16, 8, 12, 20, 24, 8, 24, 8, 16, 20, 8, 20, 0, 16, 24, 18, 10, 8, 24, 12, 32, 24, 0, 16, 24, 12, 16, 20, 0, 24, 12, 8, 16, 28, 16, 16, 48, 8, 30, 32, 8, 20, 24, 16, 0, 16, 24, 18
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*q + 2*q^2 + 4*q^3 + 4*q^4 + 4*q^5 + 8*q^6 + 8*q^7 + 6*q^8 + ...

References

  • J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p 102 eq 9.

Links

Crossrefs

Cf. A004015, A008443, A014455, A033763, A045826, A045828, A045831, A045834, A213022, A213622, A213624, A213625, A246811.

Programs

  • Magma
    A := Basis( ModularForms( Gamma1(16), 3/2), 82); A[1] + 2*A[2] + 2*A[3] + 4*A[4] + 4*A[5] + 4*A[6] + 8*A[7] + 8*A[8] + 6*A[9] + 8*A[10] + 4*A[11]; /* Michael Somos, Sep 03 2014 */
  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2] EllipticTheta[ 3, 0, q^4], {q, 0, n}]; (* Michael Somos, Sep 03 2014 *)
  • PARI
    {a(n) = my(G); if( n<0, 0, G = [1, 0, 0; 0, 2, 0; 0, 0, 4]; polcoeff( 1 + 2 * x * Ser(qfrep( G, n)), n))}; /* Michael Somos, Sep 03 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) * eta(x^8 + A)^3 / (eta(x + A)^2 * eta(x^16 + A)^2), n))}; /* Michael Somos, Sep 03 2014 */

Formula

Expansion of phi(q) * phi(q^2) * phi(q^4) in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Sep 03 2014
Euler transform of period 16 sequence [2, -1, 2, -2, 2, -1, 2, -5, 2, -1, 2, -2, 2, -1, 2, -3, ...]. - Michael Somos, Sep 03 2014
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 8 (t/i)^(3/2) f(t) where q = exp(2 Pi i t). - Michael Somos, Sep 03 2014
a(2*n + 1) = 2 * A045828(n). a(4*n) = A014455(n). a(4*n + 1) = 2 * A213625(n). a(4*n + 2) = 2 * A246811(n). a(4*n + 3) = 4 * A213624(n). - Michael Somos, Sep 03 2014
a(8*n) = A005875(n). a(8*n + 1) = 2 * A213622(n). a(8*n + 2) = 2 * A045834(n). a(8*n + 7) = 8 * A033763(n). - Michael Somos, Sep 03 2014
a(16*n) = A004015(n). a(16*n + 2) = 2 * A213022(n). a(16*n + 6) = 8 *
A008443(n). a(16*n + 8) = 2 * A045826(n). a(16*n + 10) = 8 * A045831(n). a(16*n + 14) = 0. - Michael Somos, Sep 03 2014
G.f.: theta_3(q) * theta_3(q^2) * theta_3(q^4).

A246815 Expansion of phi(-x) * psi(-x^2)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, -2, -2, 4, 3, -2, -6, 4, 4, -6, -4, 4, 7, -8, -2, 8, 8, -4, -10, 4, 4, -10, -10, 8, 9, -4, -6, 12, 8, -6, -10, 12, 4, -14, -8, 4, 16, -10, -8, 8, 9, -10, -12, 12, 8, -12, -12, 4, 20, -10, -6, 20, 8, -6, -10, 12, 8, -20, -18, 8, 11, -12, -12, 16, 8, -6, -20
Offset: 0

Views

Author

Michael Somos, Sep 03 2014

Keywords

Comments

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

Examples

			G.f. = 1 - 2*x - 2*x^2 + 4*x^3 + 3*x^4 - 2*x^5 - 6*x^6 + 4*x^7 + 4*x^8 + ...
G.f. = q - 2*q^3 - 2*q^5 + 4*q^7 + 3*q^9 - 2*q^11 - 6*q^13 + 4*q^15 + ...

Links

Crossrefs

Cf. A033763, A045828, A213622.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] EllipticTheta[ 2, Pi/4, x]^2 / (2 x^(1/2)), {x, 0, n}];
  • 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)^2, n))};

Formula

Expansion of q^(-1/2) * eta(q)^2 * eta(q^2) * eta(q^8)^2 / eta(q^4)^2 in powers of q
Euler transform of period 8 sequence [ -2, -3, -2, -1, -2, -3, -2, -3, ...].
a(n) = (-1)^floor((n+1) / 2) * A045828(n).
a(4*n) = A213622(n). a(4*n + 3) = 4 * A033763(n).

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

A246835 Expansion of psi(-x)^2 * phi(x^2) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, -2, 3, -6, 4, -4, 7, -2, 8, -10, 4, -10, 9, -6, 8, -10, 4, -8, 16, -8, 9, -12, 8, -12, 20, -6, 8, -10, 8, -18, 11, -12, 8, -20, 12, -8, 20, -6, 20, -26, 8, -8, 15, -10, 16, -18, 12, -16, 20, -10, 16, -16, 8, -24, 24, -8, 21, -26, 8, -20, 20, -14, 8, -28
Offset: 0

Views

Author

Michael Somos, Sep 04 2014

Keywords

Comments

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

Examples

			G.f. = 1 - 2*x + 3*x^2 - 6*x^3 + 4*x^4 - 4*x^5 + 7*x^6 - 2*x^7 + 8*x^8 + ...
G.f. = q - 2*q^5 + 3*q^9 - 6*q^13 + 4*q^17 - 4*q^21 + 7*q^25 - 2*q^29 + ...

Links

Crossrefs

Cf. A132969, A213622, A213625, A246836.

Programs

  • Mathematica
    a[n_]:= SeriesCoefficient[EllipticTheta[3, 0, q^2]* EllipticTheta[2, 0, I*q^(1/2)]^2/(4*(-q)^(1/4)), {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^4 + A)^7 / (eta(x^2 + A)^4 * eta(x^8 + A)^2), n))};

Formula

Expansion of q^(-1/4) * eta(q)^2 * eta(q^4)^7 / (eta(q^2)^4 * eta(q^8)^2) in powers of q.
Euler transform of period 8 sequence [ -2, 2, -2, -5, -2, 2, -2, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 16 (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246836.
a(n) = (-1)^n * A213625(n). a(2*n) = A213622(n). a(2*n + 1) = -2 * A132969(n).
Showing 1-5 of 5 results.

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

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