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 1-4 of 4 results.

A233698 Expansion of b(q^2) * c(q^2) / (3 * b(q)^2) in powers of q where b(), c() are cubic AGM functions.

Original entry on oeis.org

1, 6, 25, 84, 248, 666, 1662, 3912, 8774, 18894, 39289, 79248, 155612, 298338, 559812, 1030224, 1862647, 3313494, 5807096, 10037796, 17129888, 28886052, 48170178, 79492824, 129900206, 210314976, 337545438, 537278124, 848509124, 1330069554, 2070183912
Offset: 0

Views

Author

Michael Somos, Dec 14 2013

Keywords

Comments

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Examples

			G.f. = 1 + 6*x + 25*x^2 + 84*x^3 + 248*x^4 + 666*x^5 + 1662*x^6 + 3912*x^7 + ...
G.f. = q^2 + 6*q^5 + 25*q^8 + 84*q^11 + 248*q^14 + 666*q^17 + 1662*q^20 + ...

Links

Crossrefs

Cf. A132977, A164271, A233670.

Programs

  • Mathematica
    nmax=60; CoefficientList[Series[Product[((1-x^(2*k)) * (1-x^(3*k)) * (1-x^(6*k)) / (1-x^k)^3)^2,{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 13 2015 *)
eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q^(-2/3) *(eta[q^2]*eta[q^3]*eta[q^6]/eta[q]^3)^2, {q, 0, 50}], q] (* G. C. Greubel, Aug 07 2018 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A) * eta(x^6 + A) / eta(x + A)^3)^2, n))}

Formula

Expansion of (eta(q^2) * eta(q^3) * eta(q^6) / eta(q)^3)^2 in powers of q.
Euler transform of period 6 sequence [ 6, 4, 4, 4, 6, 0, ...].
a(n) = (-1)^n * A164271(n). 2 * a(n) = A132977(2*n + 1). -3 * a(n) = A233670(6*n + 4).
a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (2^(11/4) * 3^(9/4) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015

A261325 Expansion of f(x^3, x^3) * f(x, x^5) / f(x, x)^2 in powers of x where f(,) is Ramanujan's general theta function.

Original entry on oeis.org

1, -3, 8, -18, 38, -75, 140, -252, 439, -744, 1232, -1998, 3182, -4986, 7700, -11736, 17673, -26322, 38808, -56682, 82070, -117867, 167996, -237744, 334202, -466836, 648224, -895014, 1229148, -1679436, 2283568, -3090672, 4164578, -5587941, 7467464, -9940482
Offset: 0

Views

Author

Michael Somos, Aug 14 2015

Keywords

Comments

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

Examples

			G.f. = 1 - 3*x + 8*x^2 - 18*x^3 + 38*x^4 - 75*x^5 + 140*x^6 - 252*x^7 + ...
G.f. = q - 3*q^4 + 8*q^7 - 18*q^10 + 38*q^13 - 75*q^16 + 140*q^19 + ...

Links

Crossrefs

Cf. A187153, A213265, A233670, A233672, A233673, A233698, A260215.

Programs

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

Formula

Expansion of f(-x^2) * f(x^3) * f(-x^6) / f(x)^3 in powers of x where f() is a Ramanujan theta function.
Expansion of q^(-1/3) * eta(q)^3 * eta(q^4)^3 * eta(q^6)^4 / (eta(q^2)^8 * eta(q^3) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ -3, 5, -2, 2, -3, 2, -3, 2, -2, 5, -3, 0, ...].
a(n) = A187153(3*n + 1) = A213265(3*n + 1) = A233670(3*n + 1) = A233672(3*n + 1).
2 * a(n) = A233673(3*n + 1) = - A260215(3*n + 1). a(2*n + 1) = -3 * A233698(n).
a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/3)) / (4*3^(5/4)*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017

A233673 Expansion of phi(q) * phi(q^9) / phi(q^3)^2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 2, 0, -4, -6, 0, 12, 16, 0, -28, -36, 0, 60, 76, 0, -120, -150, 0, 228, 280, 0, -416, -504, 0, 732, 878, 0, -1252, -1488, 0, 2088, 2464, 0, -3408, -3996, 0, 5460, 6364, 0, -8600, -9972, 0, 13344, 15400, 0, -20424, -23472, 0, 30876, 35346, 0, -46152, -52644
Offset: 0

Views

Author

Michael Somos, Dec 14 2013

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^3 - 6*q^4 + 12*q^6 + 16*q^7 - 28*q^9 - 36*q^10 + ...

Links

Crossrefs

Cf. A233670.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^9] / EllipticTheta[ 3, 0, q^3]^2, {q, 0, n}]; (* Michael Somos, Aug 27 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A)^4 * eta(x^12 + A)^4 * eta(x^18 + A)^5 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^6 + A)^10 * eta(x^9 + A)^2 * eta(x^36 + A)^2), n))};

Formula

Expansion of eta(q^2)^5 * eta(q^3)^4 * eta(q^12)^4 * eta(q^18)^5 / (eta(q)^2 * eta(q^4)^2 * eta(q^6)^10 * eta(q^9)^2 * eta(q^36)^2) in powers of q.
Euler transform of period 36 sequence [ 2, -3, -2, -1, 2, 3, 2, -1, 0, -3, 2, 1, 2, -3, -2, -1, 2, 0, 2, -1, -2, -3, 2, 1, 2, -3, 0, -1, 2, 3, 2, -1, -2, -3, 2, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = f(t) where q = exp(2 Pi i t).
a(n) = 2 * A233670(n) unless n=0.

A233672 Expansion of psi(q) * phi(-q^18) * f(-q^6) / f(q^3)^3 in powers of q where phi(), psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 0, -2, -3, 0, 6, 8, 0, -14, -18, 0, 30, 38, 0, -60, -75, 0, 114, 140, 0, -208, -252, 0, 366, 439, 0, -626, -744, 0, 1044, 1232, 0, -1704, -1998, 0, 2730, 3182, 0, -4300, -4986, 0, 6672, 7700, 0, -10212, -11736, 0, 15438, 17673, 0, -23076, -26322, 0
Offset: 0

Views

Author

Michael Somos, Dec 14 2013

Keywords

Comments

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

Examples

			G.f. = 1 + q - 2*q^3 - 3*q^4 + 6*q^6 + 8*q^7 - 14*q^9 - 18*q^10 + ...

Links

Crossrefs

Cf. A233670.

Programs

  • Mathematica
    eta[q_] := q^(1/24)*QPochhammer[q]; CoefficientList[Series[eta[q^2]^2 *eta[q^3]^3*eta[q^12]^3*eta[q^18]^2/(eta[q]*eta[q^6]^8*eta[q^36]), {q, 0, 50}], q] (* G. C. Greubel, Aug 08 2018 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A)^3 * eta(x^12 + A)^3 * eta(x^18 + A)^2 / (eta(x + A) * eta(x^6 + A)^8 * eta(x^36 + A)), n))};

Formula

Expansion of eta(q^2)^2 * eta(q^3)^3 * eta(q^12)^3 * eta(q^18)^2 / (eta(q) * eta(q^6)^8 * eta(q^36)) in powers of q.
Euler transform of period 36 sequence [ 1, -1, -2, -1, 1, 4, 1, -1, -2, -1, 1, 1, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 1, 1, -1, -2, -1, 1, 4, 1, -1, -2, -1, 1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = f(t) where q = exp(2 Pi i t).
a(n) = A233670(n) unless n=0.
Showing 1-4 of 4 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.