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.

A208274 Expansion of phi(q) / phi(q^4) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 2, 0, 0, 0, -4, 0, 0, 0, 10, 0, 0, 0, -20, 0, 0, 0, 36, 0, 0, 0, -64, 0, 0, 0, 110, 0, 0, 0, -180, 0, 0, 0, 288, 0, 0, 0, -452, 0, 0, 0, 692, 0, 0, 0, -1044, 0, 0, 0, 1554, 0, 0, 0, -2276, 0, 0, 0, 3296, 0, 0, 0, -4724, 0, 0, 0, 6696, 0, 0, 0, -9408, 0, 0
Offset: 0

Views

Author

Michael Somos, Mar 12 2012

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Differs from A127391 only at n=0. - R. J. Mathar, Mar 18 2012

Examples

			1 + 2*q - 4*q^5 + 10*q^9 - 20*q^13 + 36*q^17 - 64*q^21 + 110*q^25 - 180*q^29 + ...

Links

Crossrefs

Cf. A079006, A112128, A134746, A208604.

Programs

  • Mathematica
    a[n_]:= SeriesCoefficient[EllipticTheta[3, 0, q]/EllipticTheta[3, 0, q^4], {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Dec 04 2017 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^16 + A)^2 / (eta(x + A)^2 * eta(x^8 + A)^5), n))}

Formula

Expansion of eta(q^2)^5 * eta(q^16)^2 / (eta(q)^2 * eta(q^8)^5) in powers of q.
Euler transform of period 16 sequence [ 2, -3, 2, -3, 2, -3, 2, 2, 2, -3, 2, -3, 2, -3, 2, 0, ...].
G.f. A(x) satisfies A(x)^2 - 2*A(x) + 2 = A134746(x^2), which means (phi(q) / phi(q^4) - 1)^2 + 1 = (phi(q^2) / phi(q^4))^2.
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u^2 - 2*u + 2) * (v^2 - 2*v + 2) - v^2.
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = 4 * u * (u - 1) * (2 - u) * v * (v - 1) * (2 - v) - (u - v)^4.
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = g(t) where q = exp(2 Pi i t) and g() is g.f. for A112128.
a(4*n) = 0 unless n=0. a(4*n + 2) = a(4*n + 3) = 0. a(4*n + 1) = 2 * A079006(n). a(n) = (-1)^n * A208604(n). Convolution inverse is A112128.

A210066 Expansion of (phi(q^2) / phi(q))^2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -4, 16, -48, 128, -312, 704, -1504, 3072, -6036, 11488, -21264, 38400, -67864, 117632, -200352, 335872, -554952, 904784, -1457136, 2320128, -3655296, 5702208, -8813472, 13504512, -20523996, 30952544, -46340832, 68901888, -101777112, 149403264, -218016640
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^2 - 48*q^3 + 128*q^4 - 312*q^5 + 704*q^6 - 1504*q^7 + 3072*q^8 + ...

Links

Crossrefs

Cf. A131126, A134746, A210065.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[Product[((1 - x^k) / (1 - x^(8*k)))^4 * (1 + x^(2*k))^14, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 17 2017 *)
a[n_] := SeriesCoefficient[(EllipticTheta[3, 0, q^2]/ EllipticTheta[3, 0, q])^2, {q, 0, n}]; Table[a[n], {n, 0, 30}] (* G. C. Greubel, Nov 29 2017 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( (eta(x + A) / eta(x^8 + A))^2 * (eta(x^4 + A) / eta(x^2 + A))^7)^2, n))}

Formula

Expansion of ( (eta(q) / eta(q^8))^2 * (eta(q^4) / eta(q^2))^7)^2 in powers of q.
Euler transform of period 8 sequence [ -4, 10, -4, -4, -4, 10, -4, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = (1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A134746.
a(n) = (-1)^n * A131126(n). Convolution inverse of A134746. Convolution square of A210065.
a(n) ~ (-1)^n * exp(sqrt(2*n)*Pi) / (2^(17/4) * n^(3/4)). - Vaclav Kotesovec, Nov 17 2017
Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = 1/2 + sqrt(-4+3*sqrt(2)). - Simon Plouffe, Mar 02 2021

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.

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