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.

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

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

A298733 Expansion of phi(-x^9) * f(-x^3)^2 / f(-x^2)^3 in powers of x where f(), phi() are Ramanujan theta functions.

Original entry on oeis.org

1, 0, 3, -2, 9, -6, 21, -18, 48, -44, 99, -102, 204, -216, 393, -438, 747, -846, 1362, -1584, 2448, -2872, 4275, -5082, 7356, -8784, 12390, -14894, 20592, -24798, 33651, -40644, 54336, -65640, 86535, -104628, 136356, -164736, 212388, -256498, 327690, -395214
Offset: 0

Views

Author

Michael Somos, Jan 29 2018

Keywords

Comments

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

Examples

			G.f. = 1 + 3*x^2 - 2*x^3 + 9*x^4 - 6*x^5 + 21*x^6 - 18*x^7 + 48*x^8 + ...

Links

Crossrefs

Cf. A182036.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^9] QPochhammer[ x^3]^2 / QPochhammer[ x^2]^3, {x, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^2 * eta(x^9 + A)^2 / (eta(x^2 + A)^3 * eta(x^18 + A)), n))};
  • PARI
    lista(nn) = {q='q+O('q^nn); Vec(eta(q^3)^2*eta(q^9)^2/(eta(q^2)^3*eta(q^18)))} \\ Altug Alkan, Mar 21 2018

Formula

Expansion of eta(q^3)^2 * eta(q^9)^2 / (eta(q^2)^3 * eta(q^18)) in powers of q.
Euler transform of period 18 sequence [0, 3, -2, 3, 0, 1, 0, 3, -4, 3, 0, 1, 0, 3, -2, 3, 0, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 4/9 g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A182036.
a(n) ~ (-1)^n * exp(2^(3/2)*Pi*sqrt(n)/3) / (2^(3/4) * 3^(5/2) * n^(3/4)). - Vaclav Kotesovec, Mar 21 2018
Showing 1-3 of 3 results.

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