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.

A132179 Expansion of f(-x^2)^2 * f(x, x^2) / f(-x^3)^3 in powers of x where f(,) is a Ramanujan theta function.

Original entry on oeis.org

1, 1, -1, 1, 0, -3, 4, 1, -6, 5, 1, -10, 11, 4, -19, 17, 4, -31, 31, 9, -50, 46, 11, -79, 77, 21, -122, 112, 28, -183, 173, 46, -273, 249, 62, -396, 370, 98, -573, 521, 130, -815, 751, 193, -1149, 1041, 261, -1599, 1461, 373, -2214, 1998, 498, -3031, 2750, 696, -4125, 3708, 923, -5567
Offset: 0

Views

Author

Michael Somos, Aug 12 2007

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 + x^3 - 3*x^5 + 4*x^6 + x^7 - 6*x^8 + 5*x^9 + x^10 + ...
G.f. = 1/q + q^5 - q^11 + q^17 - 3*q^29 + 4*q^35 + q^41 - 6*q^47 + 5*q^53 + ...

Links

Crossrefs

Cf. A058487, A062242, A062244, A092848, A132180, A132301, A182036, A230256, A233037, A254525.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^3 / (QPochhammer[ x]  QPochhammer[ x^3] QPochhammer[ x^6]), {x, 0, n}]; (* Michael Somos, Feb 05 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 / (eta(x + A) * eta(x^3 + A) * eta(x^6 + A)), n))};

Formula

Expansion of (chi(-x) / chi(-x^3)^3) * (psi(x) / psi(x^3))^2 in powers of x where chi(), psi() are Ramanujan theta functions. - Michael Somos, Feb 05 2015
Expansion of q^(1/6) * eta(q^2)^3 / ( eta(q) * eta(q^3) * eta(q^6)) in powers of q.
Euler transform of period 6 sequence [ 1, -2, 2, -2, 1, 0, ...].
Given g.f. A(x), then B(q) = A(q^6)/q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (u^2 - 3*v)^3 - 4*(u^2*v^2 - v^3)*(u^2*v^2 - 2*v^3).
G.f.: Product_{k>0} (1 + x^k)^2 / ( (1 - x^k + x^(2*k)) * (1 + x^k + x^(2*k))^2).
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132180.
Convolution of A092848 and A058487. - Michael Somos, Feb 05 2015
a(n) = (-1)^n * A254525(n) = A062242(2*n) = A062244(2*n) = A132301(2*n) = A182036(3*n). - Michael Somos, Feb 05 2015
a(2*n) = A230256(n). a(2*n + 1) = A233037(n). - Michael Somos, Feb 05 2015

A132302 Expansion of f(-x, -x^5) * f(-x^6) / f(-x)^2 in powers of x where f(, ) and f() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 3, 5, 10, 15, 26, 39, 63, 92, 140, 201, 295, 415, 591, 818, 1140, 1554, 2126, 2861, 3855, 5126, 6816, 8970, 11793, 15372, 20007, 25857, 33356, 42771, 54734, 69683, 88530, 111968, 141312, 177642, 222842, 278557, 347484, 432095, 536230, 663549, 819504
Offset: 0

Views

Author

Michael Somos, Aug 17 2007

Keywords

Comments

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

Examples

			G.f. = 1 + x + 3*x^2 + 5*x^3 + 10*x^4 + 15*x^5 + 26*x^6 + 39*x^7 + ...
G.f. = q + q^3 + 3*q^5 + 5*q^7 + 10*q^9 + 15*q^11 + 26*q^13 + 39*q^15 + ...

Links

Crossrefs

Cf. A124243, A132180, A132301, A132975, A213267.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^6] QPochhammer[ x^5, x^6] QPochhammer[ x^6]^2 / QPochhammer[ x]^2, {x, 0, n}]; (* Michael Somos, Nov 01 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^3 / (eta(x + A) * eta(x^2 + A) * eta(x^3 + A)), n))};

Formula

Expansion of q^(-1/2) * eta(q^6)^3 / (eta(q) * eta(q^2) * eta(q^3)) in powers of q.
Euler transform of period 6 sequence [ 1, 2, 2, 2, 1, 0, ...].
Given g.f. A(x), then B(q) = A(q^2) * q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (u^2 - v)^3 - 4 * v^4 * (v - 3*u^2) * (2*v - 3*u^2).
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (1/6) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132301.
a(n) = A124243(2*n + 1) = A132180(2*n + 1) = A132975(2*n + 1) = A213267(2*n + 1). - Michael Somos, Nov 01 2015
a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (2^(7/4)*3^(3/2)*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017

A230256 Expansion of f(-x) * psi(x^2) * 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, 4, -6, 1, 11, -19, 4, 31, -50, 11, 77, -122, 28, 173, -273, 62, 370, -573, 130, 751, -1149, 261, 1461, -2214, 498, 2750, -4125, 923, 5022, -7472, 1663, 8936, -13202, 2919, 15551, -22817, 5019, 26521, -38681, 8467, 44417, -64438, 14035, 73197
Offset: 0

Views

Author

Michael Somos, Oct 14 2013

Keywords

Comments

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

Examples

			G.f. = 1 - x + 4*x^3 - 6*x^4 + x^5 + 11*x^6 - 19*x^7 + 4*x^8 + 31*x^9 + ...
G.f. = q^-1 - q^11 + 4*q^35 - 6*q^47 + q^59 + 11*q^71 - 19*q^83 + 4*q^95 + ...

Links

Crossrefs

Cf. A062242, A062244, A132179, A132301, A182032, A182036, A182056.
Cf. A058531, A093073, A132976, A143840, A164268, A164612, A182033, A193261, A254372.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x] EllipticTheta[ 3, 0, x^3] QPochhammer[ x] / (2 x^(1/4) QPochhammer[ x^3]^3), {x, 0, n}]; (* Michael Somos, Jan 29 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A)^2 * eta(x^6 + A)^5 / (eta(x^2 + A) * eta(x^3 + A)^5 * eta(x^12 + A)^2), n))};

Formula

Expansion of q^(1/12) * eta(q) * eta(q^4)^2 * eta(q^6)^5 / (eta(q^2) * eta(q^3)^5 * eta(q^12)^2) in powers of q.
Euler transform of period 12 sequence [ -1, 0, 4, -2, -1, 0, -1, -2, 4, 0, -1, 0, ...].
a(n) = A132179(2*n) = A062242(4*n) = A062244(4*n) = A132301(4*n) = A182056(4*n) = A182036(6*n) = A182032(12*n - 1).
a(n) = A058531(12*n) = A093073(12*n) = A132976(12*n) = A143840(12*n) = A164268(12*n) = A164612(12*n) = A182033(12*n) = A193261(12*n). - Michael Somos, Jan 29 2015

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

A263526 Expansion of f(x, x)^2 / (f(x^3, x^3) * f(x, x^5)) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, 3, 1, -3, -1, 0, 1, 6, 0, -6, -3, -3, 4, 12, 1, -12, -6, -3, 5, 24, 1, -24, -10, -6, 11, 42, 4, -42, -19, -12, 17, 72, 4, -69, -31, -18, 31, 120, 9, -114, -50, -30, 46, 189, 11, -180, -79, -48, 77, 294, 21, -276, -122, -72, 112, 450, 28, -420, -183, -108
Offset: 0

Views

Author

Michael Somos, Oct 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 + x^2 - 3*x^3 - x^4 + x^6 + 6*x^7 - 6*x^9 - 3*x^10 + ...
G.f. = 1/q + 3*q^2 + q^5 - 3*q^8 - q^11 + q^17 + 6*q^20 - 6*q^26 + ...

Links

Crossrefs

Cf. A092848, A132179, A132301, A216046, A230256, A233034, A233037, A261325

Programs

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

Formula

Expansion of f(x)^3 / (f(-x^2) * f(x^3) * f(-x^6)) in powers of x where f() is a Ramanujan theta function.
Expansion of q^(1/3) * eta(q^2)^8 * eta(q^3) * eta(q^12) / (eta(q)^3 * eta(q^4)^3 * eta(q^6)^4) in powers of q.
Euler transform of period 12 sequence [ 3, -5, 2, -2, 3, -2, 3, -2, 2, -5, 3, 0, ...].
a(n) = (-1)^n * A132301(n). Convolution inverse of A261325.
a(2*n) = A132179(n). a(2*n + 1) = 3 * A092848(n). a(4*n) = A230256(n). a(4*n + 1) = 3 * A233034(n). a(4*n + 2) = A233037(n). a(4*n + 3) = -3 * A216046(n).
Showing 1-5 of 5 results.

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