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 1-6 of 6 results.

A045828 One fourth of theta series of cubic lattice with respect to face.

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, 16, 12, 14, 8, 12, 20, 14, 6, 12, 20, 8, 26, 12, 8, 22, 8, 12, 15
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).
Number of solutions to n = t1 + t2 + 2*t3 where t1, t2, t3 are triangular numbers. - Michael Somos, Jan 02 2006
The cubic lattice is the set of triples [a, b, c] where the entries are all integers. A face is centered at a triple where one entry is an integer and the other two are one half an odd integer. - Michael Somos, Jun 29 2012

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 + 6*x^9 + ...
G.f. = q + 2*q^3 + 2*q^5 + 4*q^7 + 3*q^9 + 2*q^11 + 6*q^13 + 4*q^15 + 4*q^17 + ...

References

  • J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 107.

Links

Crossrefs

Cf. A005877, A005884, A033761, A010054, A033763, A212885.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ 1/4 EllipticTheta[ 3, 0, x] EllipticTheta[ 2, 0, x]^2, {x, 0, n + 1/2}]; (* Michael Somos, Jun 29 2012 *)
a[ n_] := SeriesCoefficient[ 1/8 EllipticTheta[ 2, 0, x^2] EllipticTheta[ 2, 0, x]^2, {x, 0, 2 n + 1}]; (* Michael Somos, Jun 29 2012 *)
QP = QPochhammer; s = (QP[q^2]^3*QP[q^4]^2)/QP[q]^2 + O[q]^90; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 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^4 + A)^2 / eta(x + A)^2, n))}; /* Michael Somos, Oct 25 2006 */

Formula

Expansion of q^(-1/2) * (eta(q^2)^3 * eta(q^4)^2) / eta(q)^2 in powers of q. - Michael Somos, Jan 02 2006
Expansion of phi(x) * psi(x^2)^2 = psi(x)^2 * psi(x^2) = psi(x)^4 / phi(x) in powers of x where phi(), psi() are Ramanujan theta functions. - Michael Somos, Jun 29 2012
Euler transform of period 4 sequence [2, -1, 2, -3, ...]. - Michael Somos, Mar 05 2003
Convolution of A033761 and A010054. - Michael Somos, Jun 29 2012
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = (1/2)^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A212885. - Michael Somos, Sep 08 2018

Extensions

Edited by Michael Somos, Mar 05 2003

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

A246631 Number of integer solutions to x^2 + 2*y^2 + 2*z^2 = n.

Original entry on oeis.org

1, 2, 4, 8, 6, 8, 8, 0, 12, 10, 8, 24, 8, 8, 16, 0, 6, 16, 12, 24, 24, 16, 8, 0, 24, 10, 24, 32, 0, 24, 16, 0, 12, 16, 16, 48, 30, 8, 24, 0, 24, 32, 16, 24, 24, 24, 16, 0, 8, 18, 28, 48, 24, 24, 32, 0, 48, 16, 8, 72, 0, 24, 32, 0, 6, 32, 32, 24, 48, 32, 16, 0
Offset: 0

Views

Author

Michael Somos, Aug 31 2014

Keywords

Comments

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

Examples

			G.f. = 1 + 2*q + 4*q^2 + 8*q^3 + 6*q^4 + 8*q^5 + 8*q^6 + 12*q^8 + 10*q^9 + ...

Links

Crossrefs

Cf. A004015, A005875, A008443, A014455, A033717, A045826, A045828, A045831, A045834, A212885, A213022, A213624, A213625, A246811.

Programs

  • Magma
    A := Basis( ModularForms( Gamma0(8), 3/2), 80); A[1] + 2*A[2];
  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2]^2, {q, 0, n}];
  • PARI
    {a(n) = if( n<1, n==0, 2 * qfrep([ 1, 0, 0; 0, 2, 0; 0, 0, 2], n)[n])};
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^4 + A)^8 / (eta(x + A)^2 * eta(x^8 + A)^4), n))};

Formula

Theta series of quadratic form with Gram matrix [ 1, 0, 0; 0, 2, 0; 0, 0, 2 ].
Expansion of phi(q) * phi(q^2)^2 = phi(-q^4)^4 / phi(-q) in powers of q where phi() is a Ramanujan theta function.
Expansion of eta(q^2) * eta(q^4)^8 / (eta(q)^2 * eta(q^8)^4) in powers of q.
Euler transform of period 8 sequence [ 2, 1, 2, -7, 2, 1, 2, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 4 (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A014455.
G.f.: theta_3(q) * theta_3(q^2)^2.
G.f.: Product{k>0} (1 - x^(2*k)) * (1 - x^(4*k))^8 / ((1 - x^k)^2 * (1 - x^(8*k))^4).
G.f.: Product{k>0} (1 + x^(2*k)) * (1 + x^k)^2 * (1 - x^(4*k))^3 / (1 + x^(4*k))^4.
a(n) = (-1)^floor((n+1) / 2) * A212885(n) = abs(A212885(n)).
a(n) = A033717(2*n). a(2*n) = A014455(n). a(2*n + 1) = 2 * A246811(n).
a(4*n) = A005875(n). a(4*n + 1) = 2 * A045834(n). a(4*n + 2) = 4 * A045828(n).
a(8*n) = A004015(n). a(8*n + 1) = 2 * A213022(n). a(8*n + 2) = 4 * A213625(n). a(8*n + 3) = 8 * A008443(n). a(8*n + 4) = 2 * A045826(n). a(8*n + 5) = 8 * A045831(n). a(8*n + 6) = 8 * A213624(n). a(8*n + 7) = 0.

A246814 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, -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
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*q - 2*q^4 + 8*q^5 - 4*q^8 - 10*q^9 + 8*q^12 + 8*q^13 + ...

Links

Crossrefs

Cf. A005876, A116597, A212885.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^4]^2, {q, 0, n}]; Table[a[n], {n, 0, 80}]
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^4 / (eta(x^2 + A) * eta(x^8 + A)^2), n))};

Formula

Expansion of eta(q)^2 * eta(q^4)^4 / (eta(q^2) * eta(q^8)^2) in powers of q.
Euler transform of period 8 sequence [ -2, -1, -2, -5, -2, -1, -2, -3, ...].
a(n) = (-1)^(mod(n,4) = 1) * A116597(n).
a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = A212885(n). a(4*n + 1) = -(-1)^n * A005876(n).

A319078 Expansion of phi(-q) * phi(q)^2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 2, -4, -8, 6, 8, -8, 0, 12, 10, -8, -24, 8, 8, -16, 0, 6, 16, -12, -24, 24, 16, -8, 0, 24, 10, -24, -32, 0, 24, -16, 0, 12, 16, -16, -48, 30, 8, -24, 0, 24, 32, -16, -24, 24, 24, -16, 0, 8, 18, -28, -48, 24, 24, -32, 0, 48, 16, -8, -72, 0, 24, -32, 0, 6, 32
Offset: 0

Views

Author

Michael Somos, Sep 09 2018

Keywords

Comments

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

Examples

			G.f. = 1 + 2*x - 4*x^2 - 8*x^3 + 6*x^4 + 8*x^5 - 8*x^6 + 12*x^8 + ...

Links

Crossrefs

Cf. A004015, A004024, A005887, A008443, A045834, A083703, A080965, A132429, A212885, A213624.

Programs

  • Magma
    A := Basis( ModularForms( Gamma0(16), 3/2), 66); A[1] + 2*A[2] - 4*A[3] - 8*A[4];
  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0, q]^2, {q, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 4, 0, q^2]^2, {q, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^9 / (eta(x + A)^2 * eta(x^4 + A)^4), n))};

Formula

Expansion of eta(q^2)^9 / (eta(q)^2 * eta(q^4)^4) in powers of q.
Expansion of phi(q) * phi(-q^2)^2 = phi(-q^2)^4 / phi(-q) in powers of q.
Euler transform of period 4 sequence [2, -7, 2, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2^(11/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A045834.
G.f. Product_{k>0} (1 - x^k)^3 * (1 + x^k)^5 / (1 + x^(2*k))^4.
a(n) = (-1)^n * A212885(n) = A083703(2*n) = A080965(2*n).
a(4*n) = a(n) * -A132429(n + 2) where A132429 is a period 4 sequence.
a(4*n) = A005875(n). a(4*n + 1) = 2 * A045834(n). a(4*n + 2) = -4 * A045828(n).
a(8*n) = A004015(n). a(8*n + 1) = 2 * A213022(n). a(8*n + 2) = -4 * A213625(n). a(8*n + 3) = -8 * A008443(n). a(8*n + 4) = A005887(n). a(8*n + 5) = 2 * A004024(n). a(8*n + 6) = -8 * A213624(n). a(8*n + 7) = 0.
Showing 1-6 of 6 results.

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