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.

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A134077 Expansion of psi(x) * phi(-x)^3 / chi(-x^3)^3 in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

1, -5, 6, 8, -23, 12, 14, -30, 18, 20, -40, 24, 31, -77, 30, 32, -60, 48, 38, -70, 42, 44, -138, 48, 57, -90, 54, 72, -100, 60, 62, -184, 84, 68, -120, 72, 74, -155, 96, 80, -239, 84, 108, -150, 90, 112, -160, 120, 98, -276, 102, 104, -240, 108, 110, -190, 114
Offset: 0

Views

Author

Michael Somos, Oct 06 2007

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 - 5*x + 6*x^2 + 8*x^3 - 23*x^4 + 12*x^5 + 14*x^6 - 30*x^7 + 18*x^8 + ...
G.f. = q - 5*q^3 + 6*q^5 + 8*q^7 - 23*q^9 + 12*q^11 + 14*q^13 - 30*q^15 + ...

Links

Crossrefs

Cf. A000203, A098098, A118271, A124449, A131944.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (1/2) x^(-1/8) EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 4, 0, x]^3 QPochhammer[ -x^3, x^3]^3, {x, 0, n}]; (* Michael Somos, Oct 27 2015 *)
a[ n_] := SeriesCoefficient[ (1/16) x^(-1/2) (EllipticTheta[ 2, 0, x^(1/2)]^4 - 9 EllipticTheta[ 2, 0, x^(3/2)]^4), {x, 0, n}]; (* Michael Somos, Oct 27 2015 *)
  • PARI
    {a(n) = my(A); if ( n<0, 0, A = x * O(x^n) ; polcoeff( eta(x + A)^5 * eta(x^6 + A)^3 / ( eta(x^2 + A) * eta(x^3 + A)^3 ), n))};

Formula

Expansion of psi(x)^4 - 9 * x * psi(x^3)^4 in powers of x where psi() is a Ramanujan theta function.
Expansion of x^(-1/2) * (b(x)^3 * c(x^2)^2 / (3 * c(x)))^(1/2) in powers of x where b(), c() are cubic AGM functions.
Expansion of q^(-1/2) * eta(q)^5 * eta(q^6)^3 / (eta(q^2) * eta(q^3)^3) in powers of q.
Euler transform of period 6 sequence [-5, -4, -2, -4, -5, -4, ...].
a(n) = b(2*n+1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = 4 - 3^(e+1), b(p^e) = (p^(e+1) - 1)/(p - 1) if p>5.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 18 (t/i)^2 g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A124449
G.f.: Product_{k>0} (1 - x^k)^2 * (1 - x^(2*k))^2 * (1 - x^k + x^(2*k))^3.
G.f.: Sum_{k>0} k * f(x^k) - 9 * k * f(x^(3*k)) where f(x) = x * (1 - x) / ((1 + x) * (1 + x^2)).
G.f.: f(x) - 3 * f(x^2) - 9 * f(x^3) + 2 * f(x^4) + 27 * f(x^6) - 18 * f(x^12) where f() is the g.f. of A000203.
a(n) = A131944(2*n + 1) = A118271(2*n + 1). a(3*n + 2) = 6 * A098098(n).

A131943 Expansion of b(q) * b(q^2) in powers of q where b() is a cubic AGM theta function.

Original entry on oeis.org

1, -3, -3, 15, -3, -18, 15, -24, -3, 69, -18, -36, 15, -42, -24, 90, -3, -54, 69, -60, -18, 120, -36, -72, 15, -93, -42, 231, -24, -90, 90, -96, -3, 180, -54, -144, 69, -114, -60, 210, -18, -126, 120, -132, -36, 414, -72, -144, 15, -171, -93, 270, -42, -162
Offset: 0

Views

Author

Michael Somos, Jul 30 2007

Keywords

Comments

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

Examples

			G.f. = 1 - 3*q - 3*q^2 + 15*q^3 - 3*q^4 - 18*q^5 + 15*q^6 - 24*q^7 - 3*q^8 +...

References

  • N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 84, Eq. (32.65).

Crossrefs

Cf. A121443, A131944, A134077, A168611.

Programs

  • Magma
    A := Basis( ModularForms( Gamma0(6), 2), 54); A[1] - 3*A[2] - 3*A[3]; /* Michael Somos, Aug 30 2014 */
  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ q]^3 QPochhammer[ q^2]^3 / (QPochhammer[ q^3] QPochhammer[ q^6]), {q, 0, n}]; (* Michael Somos, Nov 21 2013 *)
a[ n_] := If[ n < 1, Boole[n == 0], -3 Sum[ d {0, 1, 0, -2, 0, 1}[[ Mod[ d, 6] + 1]], {d, Divisors @ n}]]; (* Michael Somos, Nov 11 2015 *)
  • PARI
    {a(n) = if( n<1, n==0, -3 * sumdiv(n, d, d * ((d%6==1) + (d%6==5) - 2 * (d%6==3))))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^2 + A))^3 / (eta(x^3 + A) * eta(x^6 + A)), n))};
  • PARI
    {a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); -3 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 1, p==3, 4 - 3^(e+1), (p^(e+1) - 1) / (p - 1) )))}; /* Michael Somos, Nov 21 2013 */
  • Sage
    A = ModularForms( Gamma0(6), 2, prec=54) . basis();  A[0] - 3*A[1] - 3*A[2]; # Michael Somos, Nov 21 2013

Formula

Expansion of eta(q)^3 * eta(q^2)^3 / (eta(q^3) * eta(q^6)) in powers of q.
Euler transform of period 6 sequence [ -3, -6, -2, -6, -3, -4, ...].
a(n) = -3 * b(n) where b() is multiplicative with b(2^e) = 1, b(3^e) = 4 - 3^(e+1), b(p^e) = (p^(e+1) - 1) / (p - 1) if p>3. - Michael Somos, Nov 21 2013
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 54 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A121443.
G.f.: Product_{k>0} ((1 - x^k) * (1 - x^(2*k)))^3 / ((1 - x^(3*k)) * (1 - x^(6*k))).
G.f.: 1 - 3 * (Sum_{k>0} (6*k - 1) * x^(6*k - 1) / (1 - x^(6*k - 1)) - 2*(6*k - 5) * x^(6*k - 3) / (1 - x^(6*k - 3)) + (6*k - 5) * x^(6*k - 5) / (1 -x^(6*k - 5))).
a(n) = a(2*n). a(n) = -3 * A131944(n) unless n=0. a(3^n) = 3 * A168611(n+1). a(2*n + 1) = -3 * A134077(n). - Michael Somos, Nov 21 2013
Showing 1-2 of 2 results.

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