cp's OEIS Frontend

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

%I A282610 #17 Sep 08 2022 08:46:18
%S A282610 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,
%T A282610 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,
%U A282610 -108,30,0,36,0,0,60,0,0,-252,3,0,96,24,0,108,-15,0
%N A282610 Expansion of b(q) * b(q^3) in powers of q where b() is a cubic AGM function.
%C A282610 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%C A282610 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.
%H A282610 Seiichi Manyama, <a href="/A282610/b282610.txt">Table of n, a(n) for n = 0..10000</a>
%F A282610 Expansion of eta(q)^3 * eta(q^3)^2 / eta(q^9) in powers of q.
%F A282610 Euler transform of period 9 sequence [-3, -3, -5, -3, -3, -5, -3, -3, -4, ...].
%F A282610 a(3*n) = A281722(n). a(3*n + 1) = -3 * A030206(n). a(3*n + 2) = 0.
%e A282610 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 + ...
%t A282610 a[ n_] := SeriesCoefficient[ QPochhammer[ q]^3 QPochhammer[ q^3]^2 / QPochhammer[ q^9], {q, 0, n}];
%o A282610 {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))};
%o A282610 (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
%o A282610 (Magma) A := Basis( ModularForms( Gamma0(27), 2), 69); A[1] - 3*A[2] + 3*A[4] + 6*A[5] - 18*A[6];
%Y A282610 Cf. A030206, A281722, A282611.
%K A282610 sign
%O A282610 0,2
%A A282610 _Michael Somos_, Feb 19 2017