A282610 Expansion of b(q) * b(q^3) in powers of q where b() is a cubic AGM function.

1, -3, 0, 3, 6, 0, -18, 3, 0, 12, 0, 0, 21, -15, 0, -36, -12, 0, 36, 21, 0, 24, 0, 0, -90, 15, 0, 12, -6, 0, 54, 12, 0, -72, 0, 0, 84, -33, 0, 42, 0, 0, -144, -24, 0, 72, 0, 0, 93, 18, 0, -108, 30, 0, 36, 0, 0, 60, 0, 0, -252, 3, 0, 96, 24, 0, 108, -15, 0

Offset

0,2

Comments

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (9*t)) = 729 (t/i)^2 g(t) where g() is the g.f. for A282611.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Formula

Expansion of eta(q)^3 * eta(q^3)^2 / eta(q^9) in powers of q.

Euler transform of period 9 sequence [-3, -3, -5, -3, -3, -5, -3, -3, -4, ...].

a(3*n) = A281722(n). a(3*n + 1) = -3 * A030206(n). a(3*n + 2) = 0.

Example

G.f. = 1 - 3*q + 3*q^3 + 6*q^4 - 18*q^6 + 3*q^7 + 12*q^9 + 21*q^12 - 15*q^13 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ q]^3 QPochhammer[ q^3]^2 / QPochhammer[ q^9], {q, 0, n}];

Prog

{a(n) = if( n<0, 0, my(A = x * O(x^n)); polcoeff( eta(x + A)^3 * eta(x^3 + A)^2 / eta(x^9 + A), n))};
(PARI) first(n)=my(q='x+O('x^(n+1))); Vec(eta(q)^3 * eta(q^3)^2 / eta(q^9)) \\ Charles R Greathouse IV, Jun 02 2017
(Magma) A := Basis( ModularForms( Gamma0(27), 2), 69); A[1] - 3*A[2] + 3*A[4] + 6*A[5] - 18*A[6];

Crossrefs

Cf. A030206, A281722, A282611.

Sequence in context: A120987 A281293 A258108 * A011076 A200278 A247734

Adjacent sequences: A282607 A282608 A282609 * A282611 A282612 A282613

Keyword

sign

Author

Michael Somos, Feb 19 2017

Status

approved