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.

A210063 Expansion of psi(x^4) / phi(x) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, -2, 4, -8, 15, -26, 44, -72, 114, -178, 272, -408, 605, -884, 1276, -1824, 2580, -3616, 5028, -6936, 9498, -12922, 17468, -23472, 31369, -41700, 55156, -72616, 95172, -124202, 161436, -209016, 269616, -346562, 443952, -566856, 721530, -915642, 1158608
Offset: 0

Views

Author

Michael Somos, Mar 16 2012

Keywords

Comments

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

Examples

			1 - 2*x + 4*x^2 - 8*x^3 + 15*x^4 - 26*x^5 + 44*x^6 - 72*x^7 + 114*x^8 + ...
q - 2*q^3 + 4*q^5 - 8*q^7 + 15*q^9 - 26*q^11 + 44*q^13 - 72*q^15 + 114*q^17 + ...

Links

Crossrefs

Cf. A187154, A208589, A210030.

Programs

  • Mathematica
    CoefficientList[Series[x^(-1/2) * EllipticTheta[2, 0, x^2] / (2*EllipticTheta[3, 0, x]), {x, 0, 50}], x] (* Vaclav Kotesovec, Nov 17 2017 *)
  • PARI
    {a(n) = local(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)^5, n))}

Formula

Expansion of q^(-1/2) * eta(q)^2 * eta(q^4) * eta(q^8)^2 / eta(q^2)^5 in powers of q.
Euler transform of period 8 sequence [ -2, 3, -2, 2, -2, 3, -2, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 8^(-1/2) * g(t) where q = exp(2 Pi i t) and g() is g.f. for A210030.
a(n) = (-1)^n * A187154(n). Convolution inverse of A208589.
a(n) ~ (-1)^n * exp(sqrt(n)*Pi) / (16*n^(3/4)). - Vaclav Kotesovec, Nov 17 2017

A080015 Expansion of theta_3(q) / theta_3(q^2) in powers of q.

Original entry on oeis.org

1, 2, -2, -4, 6, 8, -12, -16, 22, 30, -40, -52, 68, 88, -112, -144, 182, 228, -286, -356, 440, 544, -668, -816, 996, 1210, -1464, -1768, 2128, 2552, -3056, -3648, 4342, 5160, -6116, -7232, 8538, 10056, -11820, -13872, 16248, 18996, -22176, -25844, 30068
Offset: 0

Views

Author

Michael Somos, Jan 20 2003

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 + 6*q^4 + 8*q^5 - 12*q^6 - 16*q^7 + 22*q^8 + ...

References

  • B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 214 Entry 24(ii).

Links

Crossrefs

Cf. A080054, A210030.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] / EllipticTheta[ 3, 0, q^2], {q, 0, n}]; (* Michael Somos, Apr 24 2015 *)
  • PARI
    {a(n) = my(A, m); if( n<0, 0, m=1; A = 1 + 2 * x + O(x^2); while( m
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^8 + A) / eta(x + A))^2 * (eta(x^2 + A) / eta(x^4 + A))^7, n))};

Formula

Expansion of phi(q) / phi(q^2) in powers of q where phi() is a Ramanujan theta function.
Expansion of eta(q^2)^7 * eta(q^8)^2 / (eta(q)^2 * eta(q^4)^7) in powers of q.
Euler transform of period 8 sequence [ 2, -5, 2, 2, 2, -5, 2, 0, ...].
G.f.: A(x)/B(x), where A(x) = Sum_{m = -infinity..infinity} x^(m^2) and B(x) = Sum_{m = -infinity..infinity} x^(2*m^2). - Vladeta Jovovic, Mar 22 2005
Expansion of phi(x) / phi(x^2) where phi() is a Ramanujan theta function.
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = 1 + (1 - u*v)^2 - v^2. - Michael Somos, Jan 31 2006
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u^4 - v^4 + 8 * u*v - 6 * u*v * (u^2 + v^2) + 4 * (u*v)^3. - Michael Somos, Jan 31 2006
Expansion of sqrt(m) in powers of q where m is the multiplier for the second degree modular equation.
G.f.: Prod_{k>0} ((1 - x^(8*k - 2)) * (1 - x^(8*k - 6)))^5 / ((1 - x^(8*k - 1)) * (1 - x^(8*k - 3)) * (1 - x^(8*k - 4)) * (1 - x^(8*k - 5)) * (1 - x^(8*k - 7)))^2.
a(n) = (-1)^n * A210030(n). a(n) = (-1)^[n/2] * A080054(n).

A210067 Expansion of (phi(-q) / phi(q^2))^2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -4, 0, 16, 0, -56, 0, 160, 0, -404, 0, 944, 0, -2072, 0, 4320, 0, -8648, 0, 16720, 0, -31360, 0, 57312, 0, -102364, 0, 179104, 0, -307672, 0, 519808, 0, -864960, 0, 1419456, 0, -2299832, 0, 3682400, 0, -5831784, 0, 9141808, 0, -14194200, 0, 21842368, 0
Offset: 0

Views

Author

Michael Somos, Mar 16 2012

Keywords

Comments

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

Examples

			1 - 4*q + 16*q^3 - 56*q^5 + 160*q^7 - 404*q^9 + 944*q^11 - 2072*q^13 + ...

Links

Crossrefs

Cf. A001938, A127393, A131126, A134746, A210030.

Programs

  • Mathematica
    a[n_] := SeriesCoefficient[(EllipticTheta[3, 0, -q]/EllipticTheta[3, 0, q^2])^2, {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^2 + A) * eta(x^8 + A)^2 / eta(x^4 + A)^5)^2, n))}

Formula

Expansion of (eta(q)^2 * eta(q^2) * eta(q^8)^2 / eta(q^4)^5)^2 in powers of q.
Euler transform of period 8 sequence [ -4, -6, -4, 4, -4, -6, -4, 0, ...].
a(2*n) = 0 unless n=0. a(2*n + 1) = -4 * A001938(n) = -A127393(n).
a(n) = (-1)^n * A134746(n).
Convolution inverse of A131126. Convolution square of A210030.
Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = -32 - 24*sqrt(2) + 4*sqrt(140+99*sqrt(2)). - Simon Plouffe, Mar 02 2021
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.