A282611 Expansion of q^(-1/3) * c(q) * c(q^3) / 9 in powers of q where c() is a cubic AGM theta function.

0, 1, 1, 2, 1, 3, 3, 4, 4, 6, 4, 6, 3, 10, 4, 8, 7, 12, 8, 10, 7, 15, 7, 16, 9, 14, 7, 14, 12, 20, 13, 16, 13, 23, 13, 18, 12, 28, 16, 20, 16, 24, 12, 28, 17, 30, 13, 24, 20, 32, 19, 32, 16, 42, 21, 28, 19, 36, 27, 30, 21, 40, 24, 40, 19, 43, 21, 34, 28, 46

Offset

0,4

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)) = (t/i)^2 g(t) where g() is the g.f. for A282610.

Links

G. C. Greubel, Table of n, a(n) for n = 0..5000

Formula

Expansion of q^(-1/3) * eta(q^3)^2 * eta(q^9)^3 / eta(q) in powers of q.

Euler transform of period 9 sequence [1, 1, -1, 1, 1, -1, 1, 1, -4, ...].

Example

G.f. = x + x^2 + 2*x^3 + x^4 + 3*x^5 + 3*x^6 + 4*x^7 + 4*x^8 + 6*x^9 + ...
G.f. = q^4 + q^7 + 2*q^10 + q^13 + 3*q^16 + 3*q^19 + 4*q^22 + 4*q^25 + ...

Mathematica

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

Prog

(PARI) {a(n) = if( n<1, 0, n--; my(A = x * O(x^n)); polcoeff( eta(x^3 + A)^2 * eta(x^9 + A)^3 / eta(x + A), n))};
(Magma) Basis( ModularForms( Gamma0(27), 2), 210)[5];

Crossrefs

Cf. A282610.

Sequence in context: A370592 A151662 A049786 * A187498 A029137 A323054

Adjacent sequences: A282608 A282609 A282610 * A282612 A282613 A282614

Keyword

nonn

Author

Michael Somos, Feb 19 2017

Status

approved