%I #26 Jan 23 2024 02:20:16
%S 1,-3,0,7,-6,0,8,-15,0,18,-12,0,14,-24,0,31,-18,0,20,-42,0,36,-24,0,
%T 31,-42,0,56,-30,0,32,-63,0,54,-48,0,38,-60,0,90,-42,0,44,-84,0,72,
%U -48,0,57,-93,0,98,-54,0,72,-120,0,90,-60,0,62,-96,0,127,-84
%N Expansion of (eta(q) * eta(q^9))^3 / eta(q^3)^2 in powers of q.
%C Number 21 of the 74 eta-quotients listed in Table I of Martin (1996).
%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H G. C. Greubel, <a href="/A106401/b106401.txt">Table of n, a(n) for n = 1..1000</a>
%H Yves Martin, <a href="http://dx.doi.org/10.1090/S0002-9947-96-01743-6">Multiplicative eta-quotients</a>, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
%H Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>, 2016.
%F G.f.: x * Product_{k>0} (1 - x^k)^3 * (1 - x^(9*k))^3 / (1 - x^(3*k))^2.
%F Expansion of b(q) * c(q^3) / 3 in powers of q where b(), c() are cubic AGM theta functions. - _Michael Somos_, Oct 17 2006
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = -v^3 + 6 * u*v*w + 4 * u*w^2 + u^2*w.
%F Euler transform of period 9 sequence [ -3, -3, -1, -3, -3, -1, -3, -3, -4, ...].
%F a(n) is multiplicative with a(3^e) = 0 if e>0, a(p^e) = (p^(e+1) - 1) / (p - 1) if e even or p == 1 (mod 3), a(p^e) = -(p^(e+1) - 1) / (p - 1) otherwise. - Michael, Somos Oct 19 2005
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = 9 (t/i)^2 f(t) where q = exp(2 Pi i t).
%F a(3*n) = 0. a(3*n + 1) = A144614(n). a(3*n + 2) = -3 * A033686(n).
%F Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = 4*Pi^2/81 = 0.487387... . - _Amiram Eldar_, Jan 23 2024
%e G.f. = q - 3*q^2 + 7*q^4 - 6*q^5 + 8*q^7 - 15*q^8 + 18*q^10 - 12*q^11 +...
%t a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^9])^3 / QPochhammer[ q^3]^2, {q, 0, n}]; (* _Michael Somos_, May 18 2015 *)
%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^9 + A))^3 / eta(x^3 + A)^2, n))};
%o (PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==3, 0, (-1)^((p%3>1) * e) * (p^(e+1) - 1) / (p - 1))))};
%o (Magma) A := Basis( ModularForms( Gamma0(9), 2), 66); A[2] - 3*A[3]; /* _Michael Somos_, May 18 2015 */
%Y Cf. A033686, A144614.
%Y Cf. A004016, A005882, A005928.
%K sign,mult
%O 1,2
%A _Michael Somos_, May 02 2005