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 1-5 of 5 results.

A123629 Expansion of b(q^2) * c(q^6) / (b(q) * c(q^3)) in powers of q where b(), c() are cubic AGM theta functions.

Original entry on oeis.org

1, 3, 6, 11, 18, 30, 48, 75, 114, 170, 252, 366, 526, 744, 1044, 1451, 1998, 2730, 3700, 4986, 6672, 8876, 11736, 15438, 20207, 26322, 34134, 44072, 56682, 72612, 92680, 117867, 149400, 188758, 237744, 298554, 373838, 466836, 581412, 722266, 895014
Offset: 1

Views

Author

Michael Somos, Oct 03 2006

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Examples

			G.f. = q + 3*q^2 + 6*q^3 + 11*q^4 + 18*q^5 + 30*q^6 + 48*q^7 + 75*q^8 + ...

Links

Crossrefs

Cf. A123676, A128638, A233698.

Programs

  • Mathematica
    nmax=60; CoefficientList[Series[Product[(1+x^k)^3 * (1+x^(9*k))^3 / (1+x^(3*k))^2,{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 10 2015 *)
A123629[n_] := SeriesCoefficient[q*(QPochhammer[q^3]/QPochhammer[q^6])^2*(QPochhammer[q^2]*QPochhammer[q^18]/(QPochhammer[q]*QPochhammer[q^9] ))^3, {q, 0, n}]; Table[A123629[n], {n, 0, 50}] (* G. C. Greubel, Oct 09 2017 *)
  • PARI
    {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^3 + A) / eta(x^6 + A))^2 * (eta(x^2 + A) * eta(x^18 + A) / (eta(x + A) * eta(x^9 + A)))^3, n))};

Formula

Expansion of (eta(q^3) / eta(q^6))^2 * (eta(q^2) * eta(q^18) / (eta(q) * eta(q^9)))^3 in powers of q.
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v - u*(6*v + 4*v^2).
Euler transform of period 18 sequence [ 3, 0, 1, 0, 3, 0, 3, 0, 4, 0, 3, 0, 3, 0, 1, 0, 3, 0, ...].
Convolution inverse is A123676. - Michael Somos, Feb 19 2015
Expansion of q * chi(-q^3)^2 / (chi(-q) * chi(-q^9))^3 in powers of q where chi() is a Ramanujan theta function. - Michael Somos, Feb 19 2015
G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = (1/4) * g(t) where q = exp(2 Pi i t) and g() is the g.f. for A123676. - Michael Somos, Feb 19 2015
a(3*n) = 6 * A128638(n). a(3*n + 2) = 3 * A233698(n). - Michael Somos, Feb 19 2015
a(n) ~ exp(2*sqrt(2*n)*Pi/3) / (2^(11/4)*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Oct 10 2015

Extensions

Typo in xrefs corrected by Vaclav Kotesovec, Oct 10 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

A261576 Expansion of 3 * b(q^2) * c(q^2) / c(q)^2 in powers of q where b(), c() are cubic AGM theta functions.

Original entry on oeis.org

1, -2, -3, 12, -10, -18, 60, -48, -75, 228, -172, -252, 732, -524, -744, 2088, -1450, -1998, 5460, -3704, -4986, 13344, -8872, -11736, 30876, -20206, -26322, 68268, -44080, -56682, 145224, -92672, -117867, 298800, -188756, -237744, 597108, -373852, -466836
Offset: 0

Views

Author

Michael Somos, Aug 25 2015

Keywords

Comments

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 - 2*x - 3*x^2 + 12*x^3 - 10*x^4 - 18*x^5 + 60*x^6 - 48*x^7 + ...

Links

Crossrefs

Cf. A233698, A258099, A261326.

Programs

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

Formula

Expansion of f(-q^2)^4 / (f(-q^3)^2 * f(q, q^2)^2) in powers of q where f(,) is Ramanujan's general theta function.
Expansion of (eta(q) * eta(q^2) * eta(q^6) / eta(q^3)^3)^2 in powers of q.
Euler transform of period 6 sequence [ -2, -4, 4, -4, -2, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = 9/4 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A258099.
a(n) = A261326(2*n). a(3*n + 2) = -3 * A233698(n).

A261992 Expansion of psi(x) * f(-x^18)^3 / (phi(-x^3) * f(-x^3)^3) in powers of x where phi(), psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 0, 6, 5, 0, 25, 19, 0, 84, 61, 0, 248, 174, 0, 666, 455, 0, 1662, 1112, 0, 3912, 2573, 0, 8774, 5689, 0, 18894, 12102, 0, 39289, 24900, 0, 79248, 49759, 0, 155612, 96902, 0, 298338, 184408, 0, 559812, 343722, 0, 1030224, 628717, 0, 1862647, 1130418, 0
Offset: 0

Views

Author

Michael Somos, Sep 07 2015

Keywords

Comments

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

Examples

			G.f. = 1 + x + 6*x^3 + 5*x^4 + 25*x^6 + 19*x^7 + 84*x^9 + 61*x^10 + ...
G.f. = q^2 + q^3 + 6*q^5 + 5*q^6 + 25*q^8 + 19*q^9 + 84*q^11 + 61*q^12 + ...

Links

Crossrefs

Cf. A128638, A139214, A233698.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (1/2) x^(-1/8) EllipticTheta[ 2, 0, x^(1/2)] QPochhammer[ x^18]^3 / (EllipticTheta[ 4, 0, x^3] QPochhammer[ x^3]^3), {x, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^6 + A) * eta(x^18 + A)^3 / (eta(x + A) * eta(x^3 + A)^5), n))};

Formula

Expansion of q^-2 * eta(q^2)^2 * eta(q^6) * eta(q^18)^3 / (eta(q) * eta(q^3)^5) in powers of q.
3 * a(n) = A139214(2*n + 4). a(3*n) = A233698(n). a(3*n + 1) = A128638(n+1). a(3*n + 2) = 0.

A261446 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, 13183238, 17421288
Offset: 0

Views

Author

Michael Somos, Aug 19 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. A058647, A123649, A132301, A139213, A139214, A186964, A187020, A233693, A233698, A261240, A261325.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ x^3] QPochhammer[ x^6] / QPochhammer[ x]^3, {x, 0, n}];
nmax=60; CoefficientList[Series[Product[(1-x^(2*k)) * (1-x^(3*k)) * (1-x^(6*k)) / (1-x^k)^3,{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 14 2015 *)
  • PARI
    {a(n) = my(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, 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^2) * eta(q^3) * eta(q^6) / eta(q)^3 in powers of q.
Euler transform of period 6 sequence [ 3, 2, 2, 2, 3, 0, ...].
a(n) = (-1)^n * A261325(n). 2 * a(2*n) = A261240(3*n + 1). a(2*n + 1) = 3 * A233698(n).
2 * a(n) = A058647(3*n + 1) = A139213(3*n + 1) = A186964(3*n + 1) = A187020(3*n + 1).
a(n) = A123649(3*n + 1) = A139214(3*n + 1) = A233693(3*n + 1).
Convolution inverse is A132301.
a(n) ~ exp(2*Pi*sqrt(n/3)) / (4 * 3^(5/4) * n^(3/4)). - Vaclav Kotesovec, Oct 14 2015
Showing 1-5 of 5 results.

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