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A282611 Expansion of q^(-1/3) * c(q) * c(q^3) / 9 in powers of q where c() is a cubic AGM theta function. 2
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 (list; graph; refs; listen; history; text; internal format)
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: A123621 A151662 A049786 * A187498 A029137 A323054

Adjacent sequences:  A282608 A282609 A282610 * A282612 A282613 A282614

KEYWORD

nonn

AUTHOR

Michael Somos, Feb 19 2017

STATUS

approved

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Last modified June 13 11:35 EDT 2021. Contains 344990 sequences. (Running on oeis4.)