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.

A229615 Expansion of q^2 * psi(q^3)^6 / psi(q)^2 in powers of q where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, 3, 0, -1, 0, 7, -8, 6, 0, 1, 0, 8, -12, 15, 0, -7, 0, 18, -16, 12, 0, 5, 0, 14, -26, 24, 0, -6, 0, 31, -24, 18, 0, -5, 0, 20, -28, 42, 0, -8, 0, 36, -48, 24, 0, 13, 0, 31, -36, 42, 0, -25, 0, 56, -40, 30, 0, 6, 0, 32, -64, 63, 0, -12, 0, 54, -48, 48, 0
Offset: 2

Views

Author

Michael Somos, Sep 26 2013

Keywords

Comments

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

Examples

			G.f. = q^2 - 2*q^3 + 3*q^4 - q^6 + 7*q^8 - 8*q^9 + 6*q^10 + q^12 + ...

Links

Crossrefs

Cf. A008438, A033686, A093829, A144614, A229616.

Programs

  • Magma
    Basis( ModularForms( Gamma0(6), 2), 70)[3] /* Michael Somos, Mar 05 2023 */
  • Mathematica
    a[ n_] := If[n < 1, 0, Sum[ {0, 1, -2, 1, 0, 0}[[ Mod[d, 6, 1]]] n/d, {d, Divisors[n]}]];
a[ n_] := If[n < 1, 0, Sum[ {0, 1/2, -2/3, 1/2, 0, 0}[[ Mod[d, 6, 1]]] d, {d, Divisors[n]}]];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q^(3/2)]^6 / EllipticTheta[ 2, 0, q^(1/2)]^2 / 16, {q, 0, n}];
  • PARI
    {a(n) = if( n<1, 0, sumdiv( n, d, n/d * [0, 0, 1, -2, 1, 0][d%6 + 1]))};
  • PARI
    {a(n) = if( n<1, 0, sumdiv( n, d, d * [0, 0, 1/2, -2/3, 1/2, 0][d%6 + 1]))};
  • PARI
    {a(n) = my(A); if( n<2, 0, n -= 2; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^6 + A)^6 / (eta(x^2 + A)^2 * eta(x^3 + A)^3))^2, n))};
  • Sage
    ModularForms( Gamma0(6), 2, prec=70).2;

Formula

Expansion of (a(q) - a(q^2))^2 / 36 = c(q^2)^4 / (9 * c(q)^2) in powers of q where a(), c() are cubic AGM theta functions.
Expansion of ((eta(q) * eta(q^6)^6) / (eta(q^2)^2 * eta(q^3)^3))^2 in powers of q.
Euler transform of period 6 sequence [ -2, 2, 4, 2, -2, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (1/12) (t / i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A229616.
G.f.: sum_{k>0} x^(6*k-4) / (1 - x^(6*k-4))^2 - 2 * x^(6*k-3) / (1 - x^(6*k-3))^2 + x^(6*k-2) / (1 - x^(6*k-2))^2.
G.f.: sum_{k>0} (3*k-2) * x^(6*k-4) / (1 - x^(6*k-4)) - (4*k-2) * x^(6*k-3) / (1 - x^(6*k-3)) + (3*k-1) * x^(6*k-2) / (1 - x^(6*k-2)).
a(6*n + 1) = a(6*n + 5) = 0. a(6*n + 2) = A144614(n). a(6*n + 3) = -2 * A008438(n). a(6*n + 4) = 3 * A033686(n).
Convolution square of A093829.

A122858 Expansion of E(k) * K(k) * (2/Pi)^2 in powers of q^2 where E(), K() are complete elliptic integrals and the nome q = exp( -Pi * K(k') / K(k)).

Original entry on oeis.org

1, 8, -8, 32, -40, 48, -32, 64, -104, 104, -48, 96, -160, 112, -64, 192, -232, 144, -104, 160, -240, 256, -96, 192, -416, 248, -112, 320, -320, 240, -192, 256, -488, 384, -144, 384, -520, 304, -160, 448, -624, 336, -256, 352, -480, 624, -192, 384, -928, 456
Offset: 0

Views

Author

Michael Somos, Sep 15 2006

Keywords

Comments

Ramanujan Lambert series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).

Examples

			G.f. = 1 + 8*q - 8*q^2 + 32*q^3 - 40*q^4 + 48*q^5 - 32*q^6 + 64*q^7 - 104*q^8 + ...

Links

Crossrefs

Cf. A002129, A143336.

Programs

  • Mathematica
    a[n_] := SeriesCoefficient[8 q D[Series[EllipticTheta[2, 0, q^(1/2)], {q, 0, n + 1}], q] / Series[EllipticTheta[2, 0, q^(1/2)], {q, 0, n + 1}], {q, 0, n}] (* Sander Mack-Crane, Nov 07 2013 *)
a[ n_] := If[ n < 1, Boole[n == 0], -8 DivisorSum[ n, # (-1)^# &]]; (* Michael Somos, Jun 02 2015 *)
a[ n_] := SeriesCoefficient[ With[{f = EllipticTheta[ 2, 0, q^(1/2)]}, 8 q D[f + O[q]^(n + 1), q] / f], {q, 0, n}]; (* Michael Somos, Jun 02 2015 *)
CoefficientList[Series[(2/Pi) EllipticE[InverseEllipticNomeQ[Sqrt[q]]] EllipticTheta[3, 0, Sqrt[q]]^2, {q, 0, 40}], q] (* Jan Mangaldan, Jul 07 2020 *)
  • PARI
    {a(n) = if( n<1, n==0, -8 * sumdiv(n, d, (-1)^d * d))};

Formula

Expansion of (2 * E(k) - k'^2 * K(k)) * K(k) * (2/Pi)^2 in powers of q.
Expansion of (E(k) + k' * K(k)) * K(k) * (2/Pi)^2 / 2 in powers of q^4.
Expansion of (4 * P(q^2) - P(q)) / 3 in powers of q where P() is a Ramanujan Lambert series.
G.f.: 1 + 8 * Sum_{k>0} x^k / (1 + x^k)^2.
G.f.: 1 - 8 * Sum_{k>0} k * (-x)^k / (1 - x^k).
G.f.: 1 + 8 * Sum_{k>0} k * x^k * (1 - 3*x^k) / (1 - x^(2*k)).
a(n) = 8 * A002129(n) unless n=0. a(n) = (-1)^n * A143336(n).
Expansion of 8*q*theta_2(0,q)' / theta_2(0,q) in powers of q=exp(2*Pi*i*tau), where theta_2(z,q) is a Jacobi theta function. - Sander Mack-Crane, Nov 07 2013
Conjecture: -3 A122858(n) - A229616(n) + 4 A282031(n) = 0 for all n. - Thomas Baruchel, Jun 23 2018

A282031 Coefficients in q-expansion of (9*E_2(q^3)-E_2(q))/8.

Original entry on oeis.org

1, 3, 9, -15, 21, 18, -45, 24, 45, -69, 54, 36, -105, 42, 72, -90, 93, 54, -207, 60, 126, -120, 108, 72, -225, 93, 126, -231, 168, 90, -270, 96, 189, -180, 162, 144, -483, 114, 180, -210, 270, 126, -360, 132, 252, -414, 216, 144, -465, 171, 279, -270, 294, 162, -693, 216, 360, -300, 270, 180
Offset: 0

Views

Author

N. J. A. Sloane, Feb 19 2017

Keywords

Links

Crossrefs

See A006352 for E_2.

Formula

Conjecture: -3 A122858(n) - A229616(n) + 4 A282031(n) = 0 for all n. - Thomas Baruchel, Jun 23 2018

A321465 Expansion of (phi(x)^3 / phi(x^3))^2 in powers of x where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 12, 60, 156, 204, 72, -84, 96, 492, 588, 360, 144, 60, 168, 480, 936, 1068, 216, -516, 240, 1224, 1248, 720, 288, 348, 372, 840, 1884, 1632, 360, -504, 384, 2220, 1872, 1080, 576, -372, 456, 1200, 2184, 2952, 504, -672, 528, 2448, 3528, 1440, 576, 924, 684
Offset: 0

Views

Author

Michael Somos, Nov 11 2018

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Number 1 of the 126 eta-quotients listed in Table 1 of Williams 2012.

Examples

			G.f. = 1 + 12*x + 60*x^2 + 156*x^3 + 204*x^4 + 72*x^5 - 84*x^6 + ...

Links

Crossrefs

Cf. A113660, A229616, A321466.

Programs

  • Magma
    A := Basis( ModularForms( Gamma0(12), 2), 50); A[1] + 12*A[2] + 60*A[3] + 156*A[4] + 204*A[5];
  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x]^6 / EllipticTheta[ 3, 0, x^3]^2, {x, 0, n}];
a[ n_] := With[{s = If[ FractionalPart @ # > 0, 0, DivisorSigma[1, #]] &}, If[ n < 1, Boole[n == 0], 12 (s[n] + 2 s[n/2] + 9 s[n/3] + 4 s[n/4] - 54 s[n/6] + 36 s[n/12])]];
  • PARI
    {a(n) = my(s = x -> if(frac(x), 0, sigma(x))); if( n<1, n==0, 12*(s(n) + 2*s(n/2) + 9*s(n/3) + 4*s(n/4) - 54*s(n/6) + 36*s(n/12)))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^15 * eta(x^3 + A)^2 * eta(x^12 + A)^2 / (eta(x + A)^6 * eta(x^4 + A)^6 * eta(x^6 + A)^5))^2, n))};

Formula

Expansion of eta(q^2)^30 * eta(q^3)^4 * eta(q^12)^4 / (eta(q)^12 * eta(q^4)^12 * eta(q^6)^10) in powers of q.
Expansion of ((a(x) + 2*a(x^2) - 2*a(x^4))/3)^2 = (b(-x)^2 / b(x^2))^2 in powers of x where a(), b() are cubic AGM theta functions.
Euler transform of period 12 sequence [12, -18, 8, -6, 12, -12, 12, -6, 8, -18, 12, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 108 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A342166.
G.f.: (theta_3(0, x)^3 / theta_3(0, x^3))^2 where theta_3(0, x) is a Jacobi theta function.
G.f.: (Product_{k>0} f(x^k))^2 where f(x) := ((1 + x)^6 * (1 - x^2)^3 * (1 + x^6)^2) / ((1 + x^2)^6 * (1 - x^3) * (1 + x^3)^3).
a(n) = 12*(s(n) + 2*s(n/2) + 9*s(n/3) + 4*s(n/4) - 54*s(n/6) + 36*s(n/12)) if n>0 where s(x) = sum of divisors of x for integer x else 0.
a(n) = (-1)^n * A229616(n). Convolution square of A113660.
Showing 1-4 of 4 results.

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

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