A252650 - OEIS

%I #23 Sep 08 2022 08:46:10

%S 1,-2,-3,4,6,6,-12,-16,-3,4,36,12,-12,-28,-24,24,6,18,-12,-40,-18,32,

%T 72,24,-12,-62,-42,4,48,30,-72,-64,-3,48,108,48,-12,-76,-60,56,36,42,

%U -96,-88,-36,24,144,48,-12,-114,-93,72,84,54,-12,-144,-24,80,180

%N Expansion of (eta(q) * eta(q^2) * eta(q^3) / eta(q^6))^2 in powers of q.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

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

%H Seiichi Manyama, <a href="/A252650/b252650.txt">Table of n, a(n) for n = 0..10000</a>

%H C. Kassel and C. Reutenauer, <a href="http://arxiv.org/abs/1505.07229">On the Zeta Functions of Punctual Hilbert schemes of a Two-Dimensional Torus</a>, arXiv:1505.07229 [math.AG], 2015, see page 31 7.2(d).

%H Christian Kassel, Christophe Reutenauer, <a href="http://arxiv.org/abs/1603.06357">The Fourier expansion of eta(z)eta(2z)eta(3z)/eta(6z)</a>, arXiv:1603.06357 [math.NT], 2016.

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of f(-q)^4 * f(q, q^2)^2 / f(-q^3)^2 = f(-q)^4 * f(-q^6)^2 / f(-q, -q^5)^2 in powers of q where f() is a Ramanujan theta function.

%F Expansion of b(q) * c(q) * sqrt(b(q^2) / (3 * c(q^2))) in powers of q where b(), c() are cubic AGM theta functions.

%F Euler transform of period 6 sequence [-2, -4, -4, -4, -2, -4, ...].

%F G.f.: Product_{k>0} (1 - x^k)^4 / (1 - x^k + x^(2*k))^2.

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 1296 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A098098.

%e G.f. = 1 - 2*q - 3*q^2 + 4*q^3 + 6*q^4 + 6*q^5 - 12*q^6 - 16*q^7 - 3*q^8 + ...

%t QP = QPochhammer; s = (QP[q]*QP[q^2]*(QP[q^3]/QP[q^6]))^2 + O[q]^60; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 25 2015 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^2 + A) * eta(x^3 + A) / eta(x^6 + A))^2, n))};

%o (Magma) A := Basis( ModularForms( Gamma0(36), 2), 58); A[1] - 2*A[2] - 3*A[3] + 4*A[4] + 6*A[5] + 6*A[6] - 12*A[7] - 16*A[8] - 3*A[9] + 4*A[10] + 36*A[11] - 12*A[12];

%Y This is the square of the series in A258210.

%Y Cf. A098098.

%K sign

%O 0,2

%A _Michael Somos_, Mar 21 2015