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 1-2 of 2 results.

A118272 Expansion of q^(-2/3) * (eta(q) * eta(q^3) * eta(q^6) / eta(q^2))^2 in powers of q.

Original entry on oeis.org

1, -2, 1, -4, 8, -6, 6, -8, 14, -10, 1, -16, 20, -14, 12, -16, 31, -18, 8, -20, 32, -28, 18, -24, 38, -32, 6, -28, 44, -30, 24, -40, 57, -34, 14, -36, 72, -38, 30, -48, 62, -52, 1, -44, 68, -46, 48, -56, 74, -50, 20, -64, 80, -64, 42, -56, 108, -58, 12, -60, 112, -76, 48, -64, 98, -66, 31, -80, 104, -80, 54, -88
Offset: 0

Views

Author

Michael Somos, Apr 21 2006

Keywords

Comments

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

Examples

			G.f. = 1 - 2*x + x^2 - 4*x^3 + 8*x^4 - 6*x^5 + 6*x^6 - 8*x^7 + 14*x^8 + ...
G.f. = q^2 - 2*q^5 + q^8 - 4*q^11 + 8*q^14 - 6*q^17 + 6*q^20 - 8*q^23 + ...

Links

Crossrefs

Cf. A118271, A252651.

Programs

  • Magma
    A := Basis( ModularForms( Gamma0(36), 2), 180); A[3] - 2*A[6] + A[9]; /* Michael Somos, Mar 22 2015 */
  • Mathematica
    QP:= QPochhammer; a[n_]:= SeriesCoefficient[QP[x^3]^6/(QP[-x, x^3]* QP[-x^2, x^3]*QP[x^3])^2, {x, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Apr 15 2018 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^3 + A) * eta(x^6 + A) / eta(x^2 + A))^2, n))};
  • PARI
    q='q+O('q^99); Vec((eta(q)*eta(q^3)*eta(q^6)/eta(q^2))^2) \\ Altug Alkan, Apr 16 2018

Formula

Expansion of f(-x^3)^6 / f(x, x^2)^2 = phi(-x^3)^2 * f(-x, -x^5)^2 in powers of x where phi(), f() are Ramanujan theta functions. - Michael Somos, Mar 22 2015
Euler transform of period 6 sequence [ -2, 0, -4, 0, -2, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 16 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A252651. - Michael Somos, Mar 22 2015
G.f.: Product_{k>0} (1 - x^k)^2 * (1 - x^(3*k))^2 * (1 - x^(2*k) + x^(4*k))^2. - Michael Somos, Mar 22 2015
-3 * a(n) = A118271(3*n + 2).

A263773 Expansion of b(-q)^2 in powers of q where b() is a cubic AGM theta function.

Original entry on oeis.org

1, 6, 9, -12, -42, -18, 36, 48, 45, -12, -108, -36, 84, 84, 72, -72, -186, -54, 36, 120, 126, -96, -216, -72, 180, 186, 126, -12, -336, -90, 216, 192, 189, -144, -324, -144, 84, 228, 180, -168, -540, -126, 288, 264, 252, -72, -432, -144, 372, 342, 279, -216
Offset: 0

Views

Author

Michael Somos, Oct 27 2015

Keywords

Comments

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 + 6*x + 9*x^2 - 12*x^3 - 42*x^4 - 18*x^5 + 36*x^6 + 48*x^7 + 45*x^8 + ...

Links

Crossrefs

Cf. A134079, A226535, A262651.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ -q]^6 / QPochhammer[ -q^3]^2, {q, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^9 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^6 + A)^3))^2, n))};

Formula

Expansion of f(q)^6 / f(q^3)^2 in powers of q where f() is a Ramanujan theta function.
Expansion of (eta(q^2)^9 * eta(q^3) * eta(q^12) / (eta(q)^3 * eta(q^4)^3 * eta(q^6)^3))^2 in powers of q.
Euler transform of period 12 sequence [ 6, -12, 4, -6, 6, -8, 6, -6, 4, -12, 6, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 972 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A134079.
G.f.: Product_{k>0} (1 - (-x)^k)^6 / (1 - (-x)^(3*k))^2.
a(2*n + 1) = 6 * A252651(n). a(3*n + 2) = 9 * A134079(n).
Convolution square of A226535.
Showing 1-2 of 2 results.

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