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.

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

A139632 Expansion of chi(q) * chi(-q^5) in powers of q 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, 39, 44, 42, 41, 47, 54, 52, 49
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 + 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, A145703, A145705.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ x^5, x^10], {x, 0, n}]; (* Michael Somos, Sep 07 2015 *)
nmax = 100; CoefficientList[Series[Product[(1 + x^k) / ((1 + x^(2*k)) * (1 + x^(5*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^2 + A)^2 * eta(x^5 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^10 + A)), n))};

Formula

Expansion of q^(1/4) * eta(q^2)^2 * eta(q^5) / (eta(q) * eta(q^4) * eta(q^10)) in powers of q.
Euler transform of period 20 sequence [ 1, -1, 1, 0, 0, -1, 1, 0, 1, -1, 1, 0, 1, -1, 0, 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 A139631.
G.f.: Product_{k>0} (1 + x^k) / ((1 + x^(2*k)) * (1 + x^(5*k))).
a(n) = (-1)^floor((n + 1)/2) * A145705(n). - Michael Somos, Sep 07 2015
a(2*n) = A139631(n). a(2*n + 1) = A145703(n). - Michael Somos, Sep 07 2015
a(n) ~ exp(Pi*sqrt(n/10)) / (2^(5/4) * 5^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 08 2015

A145704 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
Offset: 0

Views

Author

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

Keywords

Comments

Denoted by "(160~b)" in Simon Norton's replicable function list.

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. A145705, A145706, A145707.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (QPochhammer[ x^8] QPochhammer[ x^10] + x QPochhammer[ x^2] QPochhammer[ x^40]) / (QPochhammer[ x^4] QPochhammer[ x^20]), {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^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

G.f. is a period 1 Fourier series which satisfies f(-1 / (1280 t)) = f(t) where q = exp(2 Pi i t).
a(n) = (-1)^n * A145705(n). a(2*n) = A145706(n). a(2*n + 1) = A145707(n).

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 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.