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.

A246833 Expansion of psi(-x)^2 * psi(x^4) in powers of x where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, 1, -2, 3, -2, 4, -4, 2, -2, 5, -4, 2, -6, 3, -6, 7, -2, 5, -4, 5, -6, 6, -2, 5, -10, 3, -6, 10, -4, 6, -8, 3, -8, 7, -6, 7, -6, 4, -6, 11, -6, 9, -10, 3, -6, 14, -4, 8, -10, 8, -10, 5, -6, 4, -16, 7, -4, 10, -4, 13, -14, 7, -8, 8, -6, 10, -12, 7, -12
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 + x^2 - 2*x^3 + 3*x^4 - 2*x^5 + 4*x^6 - 4*x^7 + 2*x^8 - 2*x^9 + ...
G.f. = q^3 - 2*q^7 + q^11 - 2*q^15 + 3*q^19 - 2*q^23 + 4*q^27 - 4*q^31 + 2*q^35 + ...

Links

Crossrefs

Cf. A033763, A213624, A246815, A246832.

Programs

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

Formula

Expansion of q^(-3/4) * eta(q)^2 * eta(q^4) * eta(q^8)^2 / eta(q^2)^2 in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 8 (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246815.
a(n) = (-1)^n * A213624(n). a(2*n) = A246832(n). a(2*n + 1) = -2 * A033763(n).

A246836 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 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 - 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 + 4*q^17 + ...

Links

Crossrefs

Cf. A045828, A246833, A246835.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, x]^2 EllipticTheta[ 3, 0, x] / (2 x^(1/2)), {x, 0, n}];
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^7 * eta(x^8 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^4), n))};

Formula

Expansion of q^(-1/2) * eta(q^2)^7 * eta(q^8)^2 / (eta(q)^2 * eta(q^4)^4) in powers of q.
Euler transform of period 8 sequence [ 2, -5, 2, -1, 2, -5, 2, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 32 (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246835.
a(n) = (-1)^floor(n/2) * A045828(n). a(n) = (-1)^n * A246815(n).
a(2*n) = A246835(n). a(2*n + 1) = 2 * A246833(n).

A246816 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
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^2 - 4*q^3 - 4*q^5 + 8*q^7 - 2*q^8 + 6*q^9 + 8*q^10 + ...

Links

Crossrefs

Cf. A127786, A246814, A246815.

Programs

  • Mathematica
    a[n_]:= SeriesCoefficient[EllipticTheta[3, 0, q]* EllipticTheta[3, 0, -q^2]*EllipticTheta[3, 0, -q^4], {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Nov 30 2017 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^7 / (eta(x + A)^2 * eta(x^4 + A) * eta(x^8 + A)), n))};

Formula

Expansion of eta(q^2)^7 / (eta(q)^2 * eta(q^4) * eta(q^8)) in powers of q.
Euler transform of period 8 sequence [ 2, -5, 2, -4, 2, -5, 2, -3, ...].
a(n) = (-1)^floor(n/2) * A127786(n). a(2*n) = A246814(n). a(2*n + 1) = 2 * A246815(n).
Showing 1-3 of 3 results.

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