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.

A139631 Expansion of chi(x^5) / chi(-x^2) in powers of x where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 2, 1, 2, 1, 3, 2, 4, 2, 5, 4, 6, 5, 8, 6, 11, 8, 13, 10, 16, 14, 20, 17, 24, 21, 31, 26, 37, 32, 44, 41, 54, 49, 64, 59, 79, 72, 94, 86, 111, 106, 132, 126, 156, 149, 187, 178, 219, 210, 257, 251, 302, 295, 352, 346, 416, 406, 483, 474, 560
Offset: 0

Views

Author

Michael Somos, Apr 27 2008

Keywords

Comments

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

Examples

			G.f. = 1 + x^2 + x^4 + x^5 + 2*x^6 + x^7 + 2*x^8 + x^9 + 3*x^10 + 2*x^11 + ...
G.f. = 1/q + q^15 + q^31 + q^39 + 2*q^47 + q^55 + 2*q^63 + q^71 + 3*q^79 + ...

Links

Crossrefs

Cf. A139632.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ -x^2, x^2] QPochhammer[ -x^5, x^10], {x, 0, n}]; (* Michael Somos, Sep 07 2015 *)
nmax = 40; CoefficientList[Series[Product[(1 + x^(2*k)) * (1 + x^(5*k)) / (1 + x^(10*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 08 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A) * eta(x^10 + A)^2 / (eta(x^2 + A) * eta(x^5 + A) * eta(x^20 + A)), n))};

Formula

Expansion of q^(1/8) * eta(q^4) * eta(q^10)^2 / (eta(q^2) * eta(q^5) * eta(q^20)) in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (640 t)) = 2^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A139632.
G.f.: Product_{k>0} (1 + x^(2*k)) * (1 + x^(5*k)) / (1 + x^(10*k)).
a(n) = A139632(2*n).
a(n) ~ exp(Pi*sqrt(n/5)) / (4 * 5^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 08 2015

A145703 Expansion of chi(x) / chi(-x^10) in powers of x where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 4, 4, 5, 6, 6, 7, 8, 10, 11, 11, 13, 15, 17, 18, 20, 23, 25, 29, 32, 34, 39, 42, 47, 52, 56, 62, 68, 77, 83, 89, 99, 108, 119, 129, 139, 154, 167, 183, 199, 214, 234, 253, 276, 299, 322, 350, 378, 413, 445, 476, 518, 559
Offset: 0

Views

Author

Michael Somos, Oct 17 2008

Keywords

Comments

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

Examples

			G.f. = 1 + x + x^3 + x^4 + x^5 + x^6 + x^7 + 2*x^8 + 2*x^9 + 3*x^10 + ...
G.f. = q^3 + q^11 + q^27 + q^35 + q^43 + q^51 + q^59 + 2*q^67 + 2*q^75 + ...

Links

Crossrefs

Cf. A139632, A145702, A145707.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Product[(1 + x^(2*k-1)) / (1 - x^(20*k-10)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 30 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ -x^10, x^10], {x, 0, n}]; (* Michael Somos, Sep 06 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^20 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^10 + A)), n))};

Formula

Expansion of q^(-3/8) * eta(q^2)^2 * eta(q^20) / (eta(q) * eta(q^4) * eta(q^10) ) in powers of q.
Euler transform of period 20 sequence [ 1, -1, 1, 0, 1, -1, 1, 0, 1, 0, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (640 t)) = 2^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A145702.
G.f.: Product_{k>0} (1 + x^(2*k - 1)) / (1 - x^(20*k - 10)).
a(n) = (-1)^n * A145707(n) = A139632(2*n + 1).
a(n) ~ exp(Pi*sqrt(n/5)) / (4*5^(1/4)*n^(3/4)). - Vaclav Kotesovec, Aug 30 2015

A145702 Expansion of chi(-x) * chi(x^5) in powers of x where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, -1, 0, -1, 1, 0, 0, -1, 1, -1, 1, -1, 2, -1, 1, -1, 2, -2, 1, -2, 3, -3, 2, -3, 4, -3, 2, -4, 5, -4, 4, -5, 6, -6, 5, -6, 8, -7, 6, -8, 11, -10, 8, -11, 13, -11, 10, -13, 16, -15, 14, -17, 20, -18, 17, -20, 24, -23, 21, -25, 31, -29, 26, -32, 37, -34, 32
Offset: 0

Views

Author

Michael Somos, Oct 17 2008

Keywords

Comments

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

Examples

			G.f. = 1 - x - x^3 + x^4 - x^7 + x^8 - x^9 + x^10 - x^11 + 2*x^12 - x^13 + ...
G.f. = 1/q - q^3 - q^11 + q^15 - q^27 + q^31 - q^35 + q^39 - q^43 + 2*q^47 + ...

Links

Crossrefs

Cf. A139631, A139632, A145703, A145704, A145705.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] QPochhammer[ -x^5, x^10], {x, 0, n}]; (* Michael Somos, Sep 06 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^10 + A)^2 / eta(x^2 + A) / eta(x^5 + A) / eta(x^20 + A), n))};

Formula

Expansion of q^(1/4) * eta(q) * eta(q^10)^2 / eta(q^2) / eta(q^5) / eta(q^20) in powers of q.
Euler transform of period 20 sequence [ -1, 0, -1, 0, 0, 0, -1, 0, -1, -1, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (640 t)) = 2^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A145703.
G.f.: Product_{k>0} (1 - x^(2*k - 1)) * (1 + x^(10*k - 5)).
a(n) = (-1)^n * A139632(n). a(2*n) = A139631(n). a(2*n + 1) = - A145703(n).
a(n) = -(-1)^floor(n/2) * A145704(n) = (-1)^floor((n + 1)/2) * A145705(n). - Michael Somos, Sep 06 2015

A262175 Expansion of chi(x) * psi(x^6) * phi(-x^30) / (f(-x^4) * psi(x^5)) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 0, 1, 2, 1, 1, 3, 4, 4, 4, 6, 8, 8, 8, 11, 16, 17, 17, 23, 31, 32, 32, 42, 54, 56, 59, 77, 94, 99, 106, 129, 156, 167, 178, 214, 257, 276, 295, 350, 416, 445, 474, 559, 652, 698, 752, 877, 1012, 1089, 1174, 1349, 1542, 1662, 1792, 2042, 2327, 2512, 2706
Offset: 0

Views

Author

Michael Somos, Sep 13 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^3 + 2*x^4 + x^5 + x^6 + 3*x^7 + 4*x^8 + 4*x^9 + ...
G.f. = q^-1 + q^11 + q^35 + 2*q^47 + q^59 + q^71 + 3*q^83 + 4*q^95 + ...

Links

Crossrefs

Cf. A139632.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ x^(-1/8) QPochhammer[ -x, x^2] EllipticTheta[ 2, 0, x^3] EllipticTheta[ 4, 0, x^30] / (QPochhammer[ x^4] EllipticTheta[ 2, 0, x^(5/2)]), {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^5 + A) * eta(x^12 + A)^2 * eta(x^30 + A)^2 / (eta(x + A) * eta(x^4 + A)^2 * eta(x^6 + A) * eta(x^10 + A)^2 * eta(x^60 + A)), n))};

Formula

Expansion of q^(1/12) * eta(q^2)^2 * eta(q^5) * eta(q^12)^2 * eta(q^30)^2 / (eta(q) * eta(q^4)^2 * eta(q^6) * eta(q^10)^2 * eta(q^60)) in powers of q.
Euler transform of a period 60 sequence.
a(n) = A139632(3*n).
a(n) ~ exp(Pi*sqrt(3*n/10)) / (2^(5/4) * 3^(3/4) * 5^(1/4) * n^(3/4)). - Vaclav Kotesovec, Nov 16 2017
Showing 1-4 of 4 results.

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