A252650 Expansion of (eta(q) * eta(q^2) * eta(q^3) / eta(q^6))^2 in powers of q.

1, -2, -3, 4, 6, 6, -12, -16, -3, 4, 36, 12, -12, -28, -24, 24, 6, 18, -12, -40, -18, 32, 72, 24, -12, -62, -42, 4, 48, 30, -72, -64, -3, 48, 108, 48, -12, -76, -60, 56, 36, 42, -96, -88, -36, 24, 144, 48, -12, -114, -93, 72, 84, 54, -12, -144, -24, 80, 180

Offset

0,2

Comments

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

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

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

C. Kassel and C. Reutenauer, On the Zeta Functions of Punctual Hilbert schemes of a Two-Dimensional Torus, arXiv:1505.07229 [math.AG], 2015, see page 31 7.2(d).

Christian Kassel, Christophe Reutenauer, The Fourier expansion of eta(z)eta(2z)eta(3z)/eta(6z), arXiv:1603.06357 [math.NT], 2016.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

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.

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.

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

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

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.

Example

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 + ...

Mathematica

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 *)

Prog

(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))};
(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];

Crossrefs

This is the square of the series in A258210.

Cf. A098098.

Sequence in context: A123131 A206398 A000793 * A265548 A062163 A002729

Adjacent sequences: A252647 A252648 A252649 * A252651 A252652 A252653

Keyword

sign

Author

Michael Somos, Mar 21 2015

Status

approved