cp's OEIS Frontend

A109039 Expansion of eta(q) * eta(q^3) * (eta(q^4) * eta(q^6) / eta(q^12))^2 in powers of q.

%I A109039 #23 Aug 06 2025 10:08:41
%S A109039 1,-1,-1,-1,-1,4,-1,6,-1,-1,4,-12,-1,-14,6,4,-1,16,-1,18,4,6,-12,-24,
%T A109039 -1,-21,-14,-1,6,28,4,30,-1,-12,16,-24,-1,-38,18,-14,4,40,6,42,-12,4,
%U A109039 -24,-48,-1,-43,-21,16,-14,52,-1,48,6,18,28,-60,4,-62,30,6
%N A109039 Expansion of eta(q) * eta(q^3) * (eta(q^4) * eta(q^6) / eta(q^12))^2 in powers of q.
%C A109039 Number 25 of the 74 eta-quotients listed in Table I of Martin (1996).
%H A109039 David Broadhurst and Daniele Dorigoni, <a href="https://arxiv.org/abs/2507.21352">Resurgent Lambert series with characters</a>, arXiv:2507.21352 [math.NT], 2025. See p. 37.
%H A109039 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 A109039 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 A109039 Euler transform of period 12 sequence [ -1, -1, -2, -3, -1, -4, -1, -3, -2, -1, -1, -4, ...].
%F A109039 G.f.: Product_{k>0} (1 - x^k) * (1 - x^(3*k)) * (1 - x^(4*k))^2 / (1 + x^(6*k))^2.
%F A109039 a(n) = -A109040(n) unless n=0. a(2*n) = a(3*n) = a(n).
%F A109039 G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 12^(3/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A124815. - _Michael Somos_, May 18 2015
%F A109039 Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(24*sqrt(3)) = 0.237425... . - _Amiram Eldar_, Jan 29 2024
%e A109039 G.f. = 1 - q - q^2 - q^3 - q^4 + 4*q^5 - q^6 + 6*q^7 - q^8 - q^9 + 4*q^10 + ...
%t A109039 a[ n_] := SeriesCoefficient[ QPochhammer[ q] QPochhammer[ q^3] (QPochhammer[ q^4] QPochhammer[ q^6] / QPochhammer[ q^12])^2, {q, 0, n}]; (* _Michael Somos_, May 18 2015 *)
%t A109039 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^3] QPochhammer[ q^3, q^6]^3 EllipticTheta[ 2, 0, q^(1/2)] EllipticTheta[ 2, Pi/4, q^(1/2)]^2 / (4 q^(3/8)), {q, 0, n}]; (* _Michael Somos_, May 18 2015 *)
%o A109039 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3 + A) * eta(x^4 + A)^2 * eta(x^6 + A)^2 / eta(x^12 + A)^2, n))};
%o A109039 (Magma) A := Basis( ModularForms( Gamma1(12), 2), 64); A[1] - A[2] - A[3] - A[4] - A[5] + 4*A[6] - A[7] + 6*A[8] - A[9]; /* _Michael Somos_, May 18 2015 */
%Y A109039 Cf. A109040, A124815.
%K A109039 sign
%O A109039 0,6
%A A109039 _Michael Somos_, Jun 17 2005