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.

A229616 Expansion of (phi(-q)^3 / phi(-q^3))^2 in powers of q 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
Offset: 0

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).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Examples

			G.f. = 1 - 12*q + 60*q^2 - 156*q^3 + 204*q^4 - 72*q^5 - 84*q^6 - 96*q^7 + ...

Links

Crossrefs

Cf. A122859, A229615.

Programs

  • Magma
    A := Basis( ModularForms( Gamma0(6), 2), 50); A[1] - 12*A[2] + 60*A[3];
  • Mathematica
    a[ n_] := If[n < 1, Boole[ n == 0], -12 Sum[ {1, -7, 10, -7, 1, 2}[[ Mod[d, 6, 1]]] n/d, {d, Divisors[n]}]];
a[ n_] := If[n < 1, Boole[ n == 0], -12 Sum[ {1, -3, 4, -3, 1, 0}[[ Mod[d, 6, 1]]] d, {d, Divisors[n]}]];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q]^6 / EllipticTheta[ 4, 0, q^3]^2, {q, 0, n}];
  • PARI
    {a(n) = if( n<1, n==0, -12 * sumdiv( n, d, n/d * [2, 1, -7, 10, -7, 1][d%6 + 1]))};
  • PARI
    {a(n) = if( n<1, n==0, -12 * sumdiv( n, d, d * [0, 1, -3, 4, -3, 1][d%6 + 1]))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^6 * eta(x^6 + A) / (eta(x^2 + A)^3 * eta(x^3 + A)^2))^2, n))};
  • Sage
    A = ModularForms( Gamma0(6), 2, prec=50).basis(); A[0] - 12*A[1] + 60*A[2];

Formula

Expansion of (2*a(q^2) - a(q))^2 = b(q)^4 / b(q^2)^2 in powers of q where a(), b() are cubic AGM theta functions.
Expansion of (eta(q)^6 * eta(q^6) / (eta(q^2)^3 * eta(q^3)^2))^2 in powers of q.
Euler transform of period 6 sequence [-12, -6, -8, -6, -12, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 432 (t / i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A229615.
G.f.: ( Product_{k>0} (1 + x^(3*k)) * (1 - x^k)^3 / ((1 + x^k)^3 * (1 - x^(3*k))))^2.
Convolution square of A122859.
Conjecture: -3 A122858(n) - A229616(n) + 4 A282031(n) = 0 for all n. - Thomas Baruchel, Jun 23 2018

A321527 Expansion of x^3 * c(x^2) * c(x^4)^2 / (9 * c(x)) in powers of x where c() is a cubic AGM theta function.

Original entry on oeis.org

0, 0, 0, 1, -1, 0, 2, 0, -3, 4, 0, 0, 1, 0, 0, 6, -7, 0, 8, 0, -6, 8, 0, 0, -1, 0, 0, 13, -8, 0, 12, 0, -15, 12, 0, 0, 7, 0, 0, 14, -18, 0, 16, 0, -12, 24, 0, 0, -5, 0, 0, 18, -14, 0, 26, 0, -24, 20, 0, 0, 6, 0, 0, 32, -31, 0, 24, 0, -18, 24, 0, 0, 5, 0, 0, 31
Offset: 0

Views

Author

Michael Somos, Nov 12 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 124 of the 126 eta-quotients listed in Table 1 of Williams 2012.

Examples

			G.f. = x^3 - x^4 + 2*x^6 - 3*x^8 + 4*x^9 + x^12 + 6*x^15 - 7*x^16 + ...

Links

Crossrefs

Cf. A008438, A033686, A097723, A112610, A144614, A224226, A229615, A321528.

Programs

  • Magma
    A := Basis( ModularForms( Gamma0(12), 2), 76); A[4] - A[5];
  • Mathematica
    a[ n_] := SeriesCoefficient[ x^3 QPochhammer[ x, x^2] QPochhammer[ x^12]^6 / (QPochhammer[ x^3, x^6]^3 QPochhammer[ x^4]^2), {x, 0, n}];
a[ n_] := With[ {s = If[ FractionalPart @ # > 0, 0, DivisorSigma[1, #]] &}, If[ n < 1, 0, s[n/3] - s[n/4] - s[n/6] + s[n/12]]];
a[ n_] := If[ n < 1, 0, Sum[ d {0, 0, 4, -3, 0, 2, 0, -3, 4, 0, 0, 0}[[Mod[d, 12, 1]]] / 12, {d, Divisors[n]}]];
  • PARI
    {a(n) = if( n<1, 0, sumdiv( n, d, d * [0, 0, 0, 4, -3, 0, 2, 0, -3, 4, 0, 0][d%12 + 1] / 12))};
  • PARI
    {a(n) = my(s = x -> if( frac(x), 0, sigma(x))); if( n<1, 0, s(n/3) - s(n/4) - s(n/6) + s(n/12))};
  • PARI
    {a(n) = my(A); n-=3; if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^3 * eta(x^12 + A)^6 / (eta(x^2 + A) * eta(x^3 + A)^3 * eta(x^4 + A)^2), n))};

Formula

Expansion of x^3 * (psi(x^3) * psi(x^6))^3 / (psi(x) * psi(x^2)) in powers of x where psi() is a Ramanujan theta function.
Expansion of x^3 * chi(-x) * f(-x^12)^6 / (chi(-x^3)^3 * f(-x^4)^2) in powers of x where chi(), f() are Ramanujan theta functions.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (1/6) (t / i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A321528.
a(n) = s(n/3) - s(n/4) - s(n/6) + s(n/12) where s(x) = sum of divisors of x for integer x else 0.
a(6*n + 1) = a(6*n + 5) = a(12*n + 2) = a(12*n + 10) = 0.
a(n) = A224226(n) if n>0. a(2*n) = -A229615(n). a(6*n + 3) = A008438(n). a(12*n + 6) = 2*A008438(n).
a(12*n + 3) = A112610(n). a(12*n + 4) = -A144614(n). a(12*n + 8) = -3*A033686(n). a(12*n + 9) = 4*A097723(n).

A232356 Expansion of 2/9 * c(q) * c(q^2) - q * (psi(q) * psi(q^3))^2 in powers of q where psi() is a Ramanujan theta function and c(q) is a cubic AGM theta function.

Original entry on oeis.org

1, 0, 5, -2, 6, 4, 8, -6, 17, 0, 12, 2, 14, 0, 30, -14, 18, 16, 20, -12, 40, 0, 24, -2, 31, 0, 53, -16, 30, 24, 32, -30, 60, 0, 48, 14, 38, 0, 70, -36, 42, 32, 44, -24, 102, 0, 48, -10, 57, 0, 90, -28, 54, 52, 72, -48, 100, 0, 60, 12, 62, 0, 136, -62, 84, 48
Offset: 1

Views

Author

Michael Somos, Nov 22 2013

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).

Examples

			G.f. = q + 5*q^3 - 2*q^4 + 6*q^5 + 4*q^6 + 8*q^7 - 6*q^8 + 17*q^9 + ...

Links

Crossrefs

Cf. A008438, A033686, A098098, A111932, A121443, A123532, A144614, A229615, A232343.

Programs

  • Magma
    Basis( ModularForms( Gamma0(6), 2), 70) [2];
  • Mathematica
    a[ n_] := If[ n < 1, 0, Sum[ d ( 2 Mod[ d, 2] Boole[Mod[ n/d, 3] > 0] - Mod[ n/d, 2] Boole[ Mod[d, 3] > 0]), {d, Divisors @n}]];
a[ n_] := SeriesCoefficient[ 2 q (QPochhammer[ q^3] QPochhammer[ q^6])^3 / (QPochhammer[ q] QPochhammer[ q^2]) - q (QPochhammer[ q^2] QPochhammer[ q^6])^4 / (QPochhammer[ q] QPochhammer[ q^3])^2, {q, 0, n}];
  • PARI
    {a(n) = local(A); if( n<1, 0, n--; A=x*O(x^n); polcoeff( 2 * (eta(x^3 + A) * eta(x^6 + A))^3 / (eta(x + A) * eta(x^2 + A)) - (eta(x^2 + A) * eta(x^6 + A))^4 / (eta(x + A) * eta(x^3 + A))^2, n))};
  • Sage
    ModularForms( Gamma0(6), 2, prec=70).1;

Formula

a(n) = 2 * A121443(n) - A111932(n). a(2*n) = -2 * A229615(n). a(12*n + 2) = a(12*n + 10) = 0.
a(n) = A123532(n) + 7 * A229615(n). a(3*n + 2) = 6 * A232343(n-1). a(6*n + 5) = 6 * A098098(n). a(12*n + 4) = -2 * A144614(n). a(12*n + 6) = 4 * A008438(n). a(12*n + 8) = -6 * A033686(n). - Michael Somos, May 23 2014

A328788 Expansion of psi(x^6)^5/psi(-x^3) * (f(-x)/f(-x^4))^3 in powers of x where psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

0, 0, 0, 1, -3, 0, 6, 0, -9, 4, 0, 0, 3, 0, 0, 6, -21, 0, 24, 0, -18, 8, 0, 0, -3, 0, 0, 13, -24, 0, 36, 0, -45, 12, 0, 0, 21, 0, 0, 14, -54, 0, 48, 0, -36, 24, 0, 0, -15, 0, 0, 18, -42, 0, 78, 0, -72, 20, 0, 0, 18, 0, 0, 32, -93, 0, 72, 0, -54, 24, 0, 0, 15
Offset: 0

Views

Author

Michael Somos, Oct 28 2019

Keywords

Comments

Number 125 of the 126 eta-quotients listed in Table 1 of Williams 2012.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 144 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A329651.

Examples

			G.f. = x^3 - 3*x^4 + 6*x^6 - 9*x^8 + 4*x^9 + 3*x^12 + 6*x^15 + ...

Links

Crossrefs

Cf. A000203, A008438, A229615, A329651.

Programs

  • Magma
    A := Basis( ModularForms( Gamma0(12), 2), 72); A[4] - 3*A[5];
  • Mathematica
    a[ n_] := SeriesCoefficient[ 2^(-9/2) x^(-15/4) (EllipticTheta[ 2, 0, x^6]^5 / EllipticTheta[ 2, Pi/4, x^3]) (QPochhammer[ x^2] / QPochhammer[ x^8])^3 ,{x,0,n}] // PowerExpand;
  • PARI
    {a(n) = my(s = x -> if(frac(x), 0, sigma(x))); if( n<3, 0, s(n/3) - 3*s(n/4) + 3*s(n/6) - s(n/12))};
  • PARI
    {a(n) = my(A); n-=3; if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^12 + A)^9 / (eta(x^3 + A) * eta(x^4 + A)^3 * eta(x^6 + A)^4), n))};

Formula

Euler transform of period 12 sequence [-3, -3, -2, 0, -3, 2, -3, 0, -2, -3, -3, -4, ...].
Expansion of phi(-x^3) * f(-x^2, -x^10)^6 / f(x, x^5)^3 in powers of x where phi(), f(,) are Ramanujan theta functions.
Expansion of eta(q)^3 * eta(q^12)^9 / (eta(q^3) * eta(q^4)^3 * eta(q^6)^4) in powers of q.
G.f.: x^3 * Product_{n>=1} (1 - x^(3*n))^4 * (1 + x^n)^2 * (1 + x^(2*n))^6 * (1 - x^n + x^(2*n))^5 * (1 - x^(2*n) + x^(4*n))^9.
a(n) = s(n/3) - 3*s(n/4) + 3*s(n/6) - s(n/12) if n>0 where s(x) = sum of divisors of x for integer x else 0.
a(2*n + 1) = -3 * A229615(n). a(6*n + 1) = a(6*n + 5) = 0. a(6*n + 3) = A008438(n).
Showing 1-4 of 4 results.

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