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.

A261320 Expansion of (phi(q^3) / phi(q))^2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -4, 12, -28, 60, -120, 228, -416, 732, -1252, 2088, -3408, 5460, -8600, 13344, -20424, 30876, -46152, 68268, -100016, 145224, -209120, 298800, -423840, 597108, -835804, 1162824, -1608508, 2212896, -3028632, 4124664, -5590976, 7544604, -10137264, 13565016
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 - 4*x + 12*x^2 - 28*x^3 + 60*x^4 - 120*x^5 + 228*x^6 + ...

Links

Crossrefs

Cf. A132002, A186924, A260215, A233673.

Programs

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

Formula

Expansion of eta(q)^4 * eta(q^4)^4 * eta(q^6)^10 / ( eta(q^2)^10 * eta(q^3)^4 * eta(q^12)^4) in powers of q.
Euler transform of period 12 sequence [ -4, 6, 0, 2, -4, 0, -4, 2, 0, 6, -4, 0, ...].
G.f.: (Sum_{k in Z} x^(3*k^2)) / (Sum_{k in Z} x^k^2)^2.
G.f.: (Product_{k>0} (1 + (-x)^k + x^(2*k)) / (1 - (-x)^k + x^(2*k)))^2.
a(n) = (-1)^n * A186924(n) = A233673(3*n) = A260215(3*n).
Convolution square of A132002.
a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/3)) / (2*3^(5/4)*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017

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

A260057 Expansion of f(-x, -x^5)^3 / (f(x, x^5) * f(-x^2, -x^2)^2) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, -4, 11, -24, 48, -92, 170, -304, 526, -884, 1451, -2336, 3700, -5772, 8876, -13472, 20207, -29988, 44072, -64184, 92680, -132760, 188758, -266512, 373838, -521152, 722266, -995432, 1364684, -1861548, 2527224, -3415344, 4595497, -6157700, 8218050, -10925848
Offset: 0

Views

Author

Michael Somos, Nov 08 2015

Keywords

Comments

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

Examples

			G.f. = 1 - 4*x + 11*x^2 - 24*x^3 + 48*x^4 - 92*x^5 + 170*x^6 - 304*x^7 + ...
G.f. = q^2 - 4*q^5 + 11*q^8 - 24*q^11 + 48*q^14 - 92*q^17 + 170*q^20 + ...

Links

Crossrefs

Cf. A260215.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ 2^(-5/2) x^(-3/4) QPochhammer[ x] / QPochhammer[ -x]^3 EllipticTheta[ 2, 0, x^(3/2)]^3 / EllipticTheta[ 2, Pi/4, x^(3/2)], {x, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^4 + A)^3 * eta(x^6 + A)^7 / (eta(x^2 + A)^9 * eta(x^3 + A)^4 * eta(x^12 + A)), n))};

Formula

Expansion of f(-x) * psi(x^3)^3 / (f(x)^3 * psi(-x^3)) in powers of x where psi(), f() are Ramanujan theta functions.
Expansion of q^(-2/3) * eta(q)^4 * eta(q^4)^3 * eta(q^6)^7 / (eta(q^2)^9 * eta(q^3)^4 * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [-4, 5, 0, 2, -4, 2, -4, 2, 0, 5, -4, 0, ...].
2 * a(n) = A260215(3*n + 2).
a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/3)) / (4*3^(5/4)*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
Showing 1-4 of 4 results.

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