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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

%I

%S 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,

%T 13,16,13,23,13,18,12,28,16,20,16,24,12,28,17,30,13,24,20,32,19,32,16,

%U 42,21,28,19,36,27,30,21,40,24,40,19,43,21,34,28,46

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

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

%C 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.

%H G. C. Greubel, <a href="/A282611/b282611.txt">Table of n, a(n) for n = 0..5000</a>

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

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

%e 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 + ...

%e G.f. = q^4 + q^7 + 2*q^10 + q^13 + 3*q^16 + 3*q^19 + 4*q^22 + 4*q^25 + ...

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

%o (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))};

%o (MAGMA) Basis( ModularForms( Gamma0(27), 2), 210)[5];

%Y Cf. A282610.

%K nonn

%O 0,4

%A _Michael Somos_, Feb 19 2017

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Last modified February 28 01:38 EST 2020. Contains 332319 sequences. (Running on oeis4.)