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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A033765 Product t2(q^d); d | 6, where t2 = theta2(q)/(2*q^(1/4)).

Original entry on oeis.org

1, 1, 1, 3, 1, 2, 5, 2, 3, 7, 4, 4, 10, 3, 3, 11, 6, 4, 12, 6, 5, 19, 6, 8, 16, 7, 10, 17, 7, 8, 25, 10, 9, 20, 8, 8, 27, 12, 11, 30, 11, 14, 27, 12, 14, 29, 14, 12, 37, 15, 11, 42, 15, 14, 34, 12, 16, 44, 18, 16, 36, 18, 17, 39, 17, 20, 59, 18, 19, 42, 22, 24, 49
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

Quadratic AGM theta functions: a(q) (see A004018), b(q) (A104794), c(q) (A005883).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Also the number of positive odd solutions to equation a^2 + 2*b^2 + 3*c^2 + 6*d^2 = 8*n + 12. - Seiichi Manyama, May 29 2017

Examples

			G.f. = 1 + x + x^2 + 3*x^3 + x^4 + 2*x^5 + 5*x^6 + 2*x^7 + 3*x^8 + 7*x^9 + ...
G.f. = q^3 + q^5 + q^7 + 3*q^9 + q^11 + 2*q^13 + 5*q^15 + 2*q^17 + 3*q^19 + ...

Links

Programs

  • Magma
    A := Basis( ModularForms( Gamma0(24), 2), 105); A[4] + A[6]; /* Michael Somos, Aug 24 2014 */
  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^2] EllipticTheta[ 2, 0, q^3] EllipticTheta[ 2, 0, q^6] / 16, {q, 0, 2 n + 3}]; (* Michael Somos, Sep 30 2013 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^4 + A)^2 * eta(x^6 + A) * eta(x^12 + A)^2 / (eta(x + A) * eta(x^3 + A)), n))}; /* Michael Somos, Sep 30 2013 */

Formula

Expansion of q^(-3) * (a(q) - a(q^3)) * c(q) / 16 in powers of q^2 where a(), c() are quadratic AGM theta functions. - Michael Somos, Sep 30 2013
Expansion of (phi(x)^2 - phi(x^3)^2) * psi(x^2)^2 / 4 in powers of x where phi(), psi() are Ramanujan theta functions. - Michael Somos, Sep 30 2013

Extensions

More terms from Seiichi Manyama, May 22 2017

A256014 Expansion of phi(-q^3)^4 / (phi(-q) * phi(-q^9)) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 2, 4, 0, -2, -8, 0, 0, 4, -4, -4, 0, 0, 4, 0, 0, -2, -8, 4, 0, 8, 0, 0, 0, 0, 6, 8, 0, 0, -8, 0, 0, 4, 0, -4, 0, 4, 4, 0, 0, -4, -8, 0, 0, 0, -8, 0, 0, 0, 2, 12, 0, -4, -8, 0, 0, 0, 0, -4, 0, 0, 4, 0, 0, -2, -16, 0, 0, 8, 0, 0, 0, 4, 4, 8, 0, 0, 0, 0, 0, 8
Offset: 0

Views

Author

Michael Somos, Jun 03 2015

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

Links

Crossrefs

Cf. A002175, A104794, A121444, A122856, A122865, A256280, A258277, A258278.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^3]^4 / (EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^9]), {q, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^8 * eta(x^18 + A) / (eta(x + A)^2 * eta(x^6 + A)^4 * eta(x^9 + A)^2), n))};
  • PARI
    {a(n) = if( n<1, n==0, 2^(n%3) * (-1)^(n\3) * sumdiv(n, d, [0, 1, 2, -1][d%4 + 1] * if(d%9, 1, 4) * (-1)^((d%8==6) + n+d)))};

Formula

Expansion of eta(q^2) * eta(q^3)^8 * eta(q^18) / (eta(q)^2 * eta(q^6)^4 * eta(q^9)^2) in powers of q.
Euler transform of period 18 sequence [ 2, 1, -6, 1, 2, -3, 2, 1, -4, 1, 2, -3, 2, 1, -6, 1, 2, -2, ...].
a(n) = (-1)^n * A256280(n). a(3*n + 1) = 2 * A258277(n). a(3*n + 2) = 4 * A258278(n). a(4*n) = A256280(n). a(4*n + 3) = a(9*n + 3) = a(9*n + 6) = 0.
a(6*n + 2) = 4 * A122865(n). a(6*n + 4) = -2 * A122856(n). a(9*n) = A104794(n). a(12*n + 1) = A002175(n). a(12*n + 5) = -8 * A121444(n).

A253185 Expansion of (phi(-q) * phi(-q^23))^2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -4, 4, 0, 4, -8, 0, 0, 4, -4, 8, 0, 0, -8, 0, 0, 4, -8, 4, 0, 8, 0, 0, -4, 16, -28, 8, -16, 32, -8, 0, -16, 20, -32, 8, 0, 36, -8, 0, -16, 40, -24, 0, -32, 0, -8, 4, -16, 64, -36, 28, -32, 40, -8, 16, -32, 32, -32, 8, -48, 32, -8, 16, -64, 52, -16, 32, 0
Offset: 0

Views

Author

Michael Somos, Mar 24 2015

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
a(n) differs from A104794(n) = (-1)^n*A004018(n) from a(23) = -4 on. - M. F. Hasler, Mar 08 2018

Examples

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

Links

Crossrefs

Cf. A104794, A253183.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (QPochhammer[ q] QPochhammer[ q^23])^4 / (QPochhammer[ q^2] QPochhammer[ q^46])^2, {q, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^23 + A))^4 / (eta(x^2 + A) * eta(x^46 + A))^2, n))};
  • PARI
    {A253185(n,o=O('x^(n+1)))= polcoeff(((eta('x+o)*eta('x^23+o))^2/(eta('x^2+o)*eta('x^46+o)))^2, n)} \\ Writing the g.f. as a square makes the code more than 2 x faster. Using 'x prevents erroneous results in case x is used elsewhere. - M. F. Hasler, Mar 08 2018
  • PARI
    A253185_vec(N)={my(q='q+O('q^N)); Vec((eta(q) * eta(q^23))^4 / (eta(q^2) * eta(q^46))^2)} \\ Joerg Arndt, Mar 09 2018

Formula

Expansion of (eta(q) * eta(q^23))^4 / (eta(q^2) * eta(q^46))^2 in powers of q.
Euler transform of a period 46 sequence.
G.f. is a period 1 Fourier series which satisfies f(-1 / (46 t)) = 368 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A253183.
G.f.: (Sum_{k in Z} q^k^2)^2 * (Sum_{k in Z} q^(23*k^2))^2.
Previous Showing 11-13 of 13 results.

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

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