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.

A261154 Expansion of psi(q^6) * f(-q^12) / (psi(-q) * psi(q^9)) in powers of q where psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 8, 11, 14, 18, 24, 30, 38, 48, 60, 75, 92, 114, 140, 170, 208, 252, 304, 366, 439, 526, 626, 744, 884, 1044, 1232, 1451, 1704, 1998, 2336, 2730, 3182, 3700, 4300, 4986, 5772, 6672, 7700, 8876, 10212, 11736, 13472, 15438, 17673, 20207
Offset: 0

Views

Author

Michael Somos, Aug 10 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 + 2*x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 11*x^8 + ...

Links

Crossrefs

Cf. A186115, A212484, A233693.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ 2^(1/2) q^(1/2) EllipticTheta[ 2, 0, q^3] QPochhammer[ q^12] / (EllipticTheta[ 2, Pi/4, q^(1/2)] EllipticTheta[ 2, 0, q^(9/2)]), {q, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^9 + A) * eta(x^12 + A)^3 / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^18 + A)^2), n))};

Formula

Expansion of eta(q^2) * eta(q^9) * eta(q^12)^3 / (eta(q) * eta(q^4) * eta(q^6) * eta(q^18)^2) in powers of q.
Euler transform of period 36 sequence [1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, -1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, -1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 1/2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A186115.
a(n) = A233693(n) unless n=0. a(2*n) = A212484(n).
a(n) ~ exp(2*Pi*sqrt(n)/3) / (4*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017

A261156 Expansion of chi(q) * chi(-q^9) / (chi(-q) * chi(q^9)) in powers of q where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, 2, 2, 4, 6, 8, 12, 16, 22, 28, 36, 48, 60, 76, 96, 120, 150, 184, 228, 280, 340, 416, 504, 608, 732, 878, 1052, 1252, 1488, 1768, 2088, 2464, 2902, 3408, 3996, 4672, 5460, 6364, 7400, 8600, 9972, 11544, 13344, 15400, 17752, 20424, 23472, 26944, 30876, 35346
Offset: 0

Views

Author

Michael Somos, Aug 10 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 + 4*x^3 + 6*x^4 + 8*x^5 + 12*x^6 + 16*x^7 + 22*x^8 + ...

Links

Crossrefs

Cf. A080054, A123629, A186924, A187100, A212484, A233693, A260215.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ -q, q^2] QPochhammer[ -q, q] QPochhammer[ q^9, q^18] QPochhammer[ q^9, -q^9], {q, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^9 + A)^2 * eta(x^36 + A) / (eta(x + A)^2 * eta(x^4 + A) * eta(x^18 + A)^3), n))};

Formula

Expansion of eta(q^2)^3 * eta(q^9)^2 * eta(q^36) / (eta(q)^2 * eta(q^4) * eta(q^18)^3) in powers of q.
G.f. A(x) = B(x) / B(x^9) where B(x) is the g.f. of A080054.
Euler transform of period 36 sequence [ 2, -1, 2, 0, 2, -1, 2, 0, 0, -1, 2, 0, 2, -1, 2, 0, 2, 0, 2, 0, 2, -1, 2, 0, 2, -1, 0, 0, 2, -1, 2, 0, 2, -1, 2, 0, ...].
a(n) = 2 * A233693(n) unless n=0. a(2*n) = 2 * A123629(n) = 2 * A212484(n) unless n=0.
a(3*n) = A186924(n). a(3*n) = 4 * A187100(n) unless n=0.
a(n) = (-1)^n * A260215(n). - Michael Somos, Aug 14 2015
a(n) ~ exp(2*Pi*sqrt(n)/3) / (2*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017

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 1-3 of 3 results.

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