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.

A145707 Expansion of chi(-q) / chi(-q^10) in powers of q 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
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).
In general, if m > 1 and g.f. = Product_{k>=1} (1 + x^(m*k))/(1 + x^k), then a(n) ~ (-1)^n * exp(Pi*sqrt((m+2)*n/(6*m))) * (m+2)^(1/4) / (4 * (6*m)^(1/4) * n^(3/4)) if m is even and a(n) ~ (-1)^n * exp(Pi*sqrt((m-1)*n/(6*m))) * (m-1)^(1/4) / (2^(3/2) * (6*m)^(1/4) * n^(3/4)) if m is odd. - Vaclav Kotesovec, Aug 31 2015

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. A145703, A145704, A145705.
Cf. A081360 (m=2), A109389 (m=3), A261734 (m=4), A133563 (m=5), A261736 (m=6), A113297 (m=7), A261735 (m=8), A261733 (m=9).

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Product[(1 + x^(10*k))/(1 + x^k), {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 + A) * eta(x^20 + A) / (eta(x^2 + A) * eta(x^10 + A)), n))};

Formula

Expansion of q^(-3/8) * eta(q) * eta(q^20) / (eta(q^2) * eta(q^10)) in powers of q.
Euler transform of period 20 sequence [ -1, 0, -1, 0, -1, 0, -1, 0, -1, 1, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (1280 t)) = f(t) where q = exp(2 Pi i t).
G.f.: Product_{k>0} (1 - x^(2*k - 1)) / (1 - x^(20*k - 10)).
a(n) = (-1)^n * A145703(n) = A145704(2*n + 1) = - A145705(2*n + 1).
a(n) ~ (-1)^n * exp(Pi*sqrt(n/5)) / (4*5^(1/4)*n^(3/4)). - Vaclav Kotesovec, Aug 30 2015

A145705 Expansion of q^(1/4) * (eta(q^8) * eta(q^10) - eta(q^2) * eta(q^40)) / (eta(q^4) * eta(q^20)) in powers of q.

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, 39, 44, -42, -41, 47, 54, -52, -49, 56, 64, -62, -59, 68, 79, -77, -72
Offset: 0

Views

Author

Michael Somos, Oct 17 2008, Nov 11 2008, Jan 21 2009

Keywords

Examples

			1/q - q^3 + q^11 + q^15 + q^27 + q^31 - q^35 - q^39 + q^43 + 2*q^47 + ...

Links

Crossrefs

(-1)^n * A145704(n) = a(n). A145706(n) = a(2*n). - A145707(n) = a(2*n + 1).

Programs

  • Mathematica
    QP = QPochhammer; s = (QP[q^8]*QP[q^10]-q*QP[q^2]*QP[q^40])/(QP[q^4]* QP[q^20]) + O[q]^90; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, adapted from PARI *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^8 + A) * eta(x^10 + A) - x * eta(x^2 + A) * eta(x^40 + A)) / (eta(x^4 + A) * eta(x^20 + A)), n))}

Formula

Denoted by "(160~a)" in Simon Norton's replicable function list.
G.f. is a period 1 Fourier series which satisfies f(-1 / (1280 t)) = f(t) where q = exp(2 Pi i t).

A145706 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
Offset: 0

Views

Author

Michael Somos, Oct 17 2008, Oct 20 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. A139631, A145704, A145705.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x^5, x^10] QPochhammer[ -x^2, x^2], {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^4 + A) * eta(x^5 + A) / (eta(x^2 + A) * eta(x^10 + A)), n))};

Formula

Expansion of q^(1/8) * eta(q^4) * eta(q^5) / (eta(q^2) * eta(q^10)) in powers of q.
Euler transform of period 20 sequence [ 0, 1, 0, 0, -1, 1, 0, 0, 0, 1, 0, 0, 0, 1, -1, 0, 0, 1, 0, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (1280 t)) = f(t) where q = exp(2 Pi i t).
G.f.: Product_{k>0} (1 - x^(10*k - 5)) / (1 - x^(4*k - 2)).
a(n) = (-1)^n * A139631(n) = A145704(2*n) = A145705(2*n).

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
Showing 1-4 of 4 results.

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