A246707 Expansion of phi(-q) * phi(-q^2) * phi(-q^3) * phi(-q^6) in powers of q.

1, -2, -2, 2, 6, 4, -14, 0, 6, -2, -12, -8, 42, 4, -16, -4, 6, -4, -50, 8, 36, 0, -24, 16, 42, 2, -28, 2, 48, -12, -84, -16, 6, 8, -36, 0, 150, -12, -40, -4, 36, 12, -112, -8, 72, 4, -48, 0, 42, 14, -62, 4, 84, 4, -158, 16, 48, -8, -60, -8, 252, 4, -64, 0, 6

Offset

0,2

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 384 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A033765.

a(2*n + 1) = -2 * A030188(n).

Example

G.f. = 1 - 2*q - 2*q^2 + 2*q^3 + 6*q^4 + 4*q^5 - 14*q^6 + 6*q^8 - 2*q^9 + ...

Mathematica

eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[eta[q]^2* eta[q^2]*eta[q^3]^2*eta[q^6]/(eta[q^4]*eta[q^12]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Apr 18 2018 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^6 + A) / (eta(x^4 + A) * eta(x^12 + A)), n))};
(PARI) q='q+O('q^99); Vec(eta(q)^2*eta(q^2)*eta(q^3)^2*eta(q^6)/(eta(q^4)*eta(q^12))) \\ Altug Alkan, Apr 18 2018
(Magma) A := Basis( ModularForms( Gamma0(24), 2), 26); A[1] - 2*A[2] - 2*A[3] + 2*A[4] + 6*A[5] + 4*A[6] - 14*A[7] + 6*A[8];

Crossrefs

Cf. A030188, A033765.

Sequence in context: A059885 A259689 A300413 * A324339 A211391 A309078

Adjacent sequences: A246704 A246705 A246706 * A246708 A246709 A246710

Keyword

sign

Author

Michael Somos, Sep 01 2014

Status

approved