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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A261118 Expansion of psi(x)^2 * psi(-x^3)^2 / (phi(-x^4) * psi(-x^6)) in power of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 2, 1, 0, 0, 2, 3, 2, 2, 0, 0, 2, 3, 2, 0, 0, 0, 0, 2, 4, 1, 0, 0, 2, 2, 2, 4, 0, 0, 0, 3, 4, 0, 0, 0, 0, 4, 2, 0, 0, 0, 4, 1, 2, 2, 0, 0, 2, 2, 2, 0, 0, 0, 0, 4, 0, 3, 0, 0, 2, 2, 6, 2, 0, 0, 2, 4, 2, 0, 0, 0, 0, 1, 2, 2, 0, 0, 2, 2, 2, 2, 0, 0, 0, 2, 4, 0
Offset: 0

Views

Author

Michael Somos, Aug 08 2015

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^5 + 3*x^6 + 2*x^7 + 2*x^8 + 2*x^11 + 3*x^12 + ...
G.f. = q + 2*q^5 + q^9 + 2*q^21 + 3*q^25 + 2*q^29 + 2*q^33 + 2*q^45 + ...

Links

Crossrefs

Cf. A129402, A190615, A192013, A259668, A259895, A260308.

Programs

  • Mathematica
    a[n_]:= SeriesCoefficient[(-1)^(-1/8)*q^(-1/4)*(EllipticTheta[2, 0, Sqrt[q]]*EllipticTheta[2, 0, I*Sqrt[q^3]])^2/(8*EllipticTheta[3, 0, -q^4]*EllipticTheta[2, 0, I*q^3]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Jan 04 2018 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^3 + A)^2 * eta(x^8 + A) * eta(x^12 + A)^3 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^6 + A)^3 * eta(x^24 + A)), n))};

Formula

Expansion of f(-x^8) * f(x, x^5)^2 / psi(-x^6) in powers of x where psi(), f() are Ramanujan theta functions.
Expansion of q^(-1/4) * eta(q^2)^4 * eta(q^3)^2 * eta(q^8) * eta(q^12)^3 / (eta(q)^2 * eta(q^4)^2 * eta(q^6)^3 * eta(q^24)) in powers of q.
Euler transform of period 24 sequence [ 2, -2, 0, 0, 2, -1, 2, -1, 0, -2, 2, -2, 2, -2, 0, -1, 2, -1, 2, 0, 0, -2, 2, -2, ...].
a(n) = (-1)^n * A259668(n) = A129402(2*n) = A190615(2*n) = A192013(4*n) = A000377(4*n + 1) = A129402(6*n + 1).
a(2*n) = A260308(n). a(2*n + 1) = 2 * A259895(n).

A261119 Expansion of f(x^2, -x^4) * f(x, x^5)^2 / f(-x^12, -x^12) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, 2, 2, 2, 0, 0, 1, 2, 4, 0, 0, 0, 0, 4, 2, 2, 0, 0, 3, 2, 2, 2, 0, 0, 2, 2, 2, 0, 0, 0, 0, 2, 2, 2, 0, 0, 3, 2, 4, 2, 0, 0, 0, 6, 2, 0, 0, 0, 0, 2, 4, 0, 0, 0, 2, 2, 2, 4, 0, 0, 1, 0, 2, 0, 0, 0, 0, 2, 6, 2, 0, 0, 2, 4, 0, 2, 0, 0, 4, 4, 0, 0, 0, 0, 0, 4, 2
Offset: 0

Views

Author

Michael Somos, Aug 08 2015

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 + 2*x^3 + x^6 + 2*x^7 + 4*x^8 + 4*x^13 + 2*x^14 + ...
G.f. = q^3 + 2*q^7 + 2*q^11 + 2*q^15 + q^27 + 2*q^31 + 4*q^35 + 4*q^55 + ...

Links

Crossrefs

Cf. A000377, A129402, A192013, A257920, A257921, A259896, A261118.

Programs

  • Mathematica
    a[ n_] := If[ n < 0, 0, With[ {m = 4 n + 3}, (-1)^n DivisorSum[ m, KroneckerSymbol[ 12, #] KroneckerSymbol[ -2, m/#] &]]]; (* Michael Somos, Dec 22 2016 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^3 + A)^2 * eta(x^4 + A) * eta(x^24 + A) / (eta(x + A)^2 * eta(x^6 + A)^2 * eta(x^8 + A)), n))};
  • PARI
    a(n) = my(m = 4*n+3); (-1)^n*sumdiv(m, d, kronecker(12, d) * kronecker(-2, m/d)); \\ Michel Marcus, Dec 13 2017

Formula

Expansion of f(x^2, x^6) * f(x, x^5)^2 / f(x^4, x^8) in powers of x where f(,) is Ramanujan'sgeneral theta function.
Expansion of q^(-3/4) * eta(q^2)^3 * eta(q^3)^2 * eta(q^4) * eta(q^24) / (eta(q)^2 * eta(q^6)^2 * eta(q^8)) in powers of q.
Euler transform of period 24 sequence [ 2, -1, 0, -2, 2, -1, 2, -1, 0, -1, 2, -2, 2, -1, 0, -1, 2, -1, 2, -2, 0, -1, 2, -2, ...].
a(n) = (-1)^n * A257921(n) = A129402(2*n + 1) = A261118(3*n + 2) = A192013(4*n + 3) = A000377(4*n + 3).
a(2*n) = A257920(n). a(2*n + 1) = 2 * A259896(n). a(3*n) = A261118(n).

A261122 Expansion of f(-x) * f(x^4, x^8)^2 / f(-x^3, -x^9) in powers of x where f(,) is Ramanujan's general theta function.

Original entry on oeis.org

1, -1, -1, 1, 1, -2, -1, 2, 1, -1, -2, 2, 1, 0, -2, 2, 1, 0, -1, 0, 2, -2, -2, 0, 1, -3, 0, 1, 2, -2, -2, 2, 1, -2, 0, 4, 1, 0, 0, 0, 2, 0, -2, 0, 2, -2, 0, 0, 1, -3, -3, 0, 0, -2, -1, 4, 2, 0, -2, 2, 2, 0, -2, 2, 1, 0, -2, 0, 0, 0, -4, 0, 1, -2, 0, 3, 0, -4
Offset: 0

Views

Author

Michael Somos, Aug 09 2015

Keywords

Comments

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

Examples

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

Links

Crossrefs

Cf. A000377, A129402, A190611, A190615, A261118, A261119.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ 2^(1/2) EllipticTheta[ 4, 0, x^12]^2 EllipticTheta[ 2, Pi/4, x]^2 / (EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, Pi/4, x^(3/2)]), {x, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A) * eta(x^8 + A)^2 * eta(x^12 + A)^3 / (eta(x^3 + A) * eta(x^4 + A)^2 * eta(x^24 + A)^2), n))};

Formula

Expansion of phi(-x^12)^2 * psi(-x^2)^2 / (psi(x) * psi(-x^3)) in powers of x where phi(), psi() are Ramanujan theta functions.
Expansion of eta(q) * eta(q^6) * eta(q^8)^2 * eta(q^12)^3 / (eta(q^3) * eta(q^4)^2 * eta(q^24)^2) in powers of q.
Euler transform of period 24 sequence [ -1, -1, 0, 1, -1, -1, -1, -1, 0, -1, -1, -2, -1, -1, 0, -1, -1, -1, -1, 1, 0, -1, -1, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (96 t)) = 384^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A261119.
a(n) = (-1)^(n + floor(n/2)) * A000377(n) = (-1)^floor(n/2) * A190611(n).
a(2*n) = A190611(n). a(2*n + 1) = - A190615(n). a(4*n) = A000377(n). a(4*n + 1) = - A261118(n). a(4*n + 2) = - A129402(n). a(4*n + 3) - A261119(n).
Previous Showing 11-13 of 13 results.

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