A123858 Expansion of q^(-1/3) * eta(q^2)^2 * eta(q^5)/eta(q) in powers of q.

1, 1, 0, 1, 0, -1, 0, 0, -1, 0, 0, -2, 0, -1, 0, 0, -1, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, -1, 0, 2, 2, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, -2, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, -2, 0, 0, 0, 2, -2, 0, -1, 0, -1, 0, 0, -2, 0, 0, 2, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0

Offset

0,12

Comments

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (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 f(-q^5)*psi(q) in powers of q where f(),psi() are Ramanujan theta functions.

Euler transform of period 10 sequence [ 1, -1, 1, -1, 0, -1, 1, -1, 1, -2, ...].

G.f.: Product_{k>0} (1+x^k)*(1-x^(2k))*(1-x^(5k)).

Mathematica

eta[q_] := q^(1/24)*QPochhammer[q]; CoefficientList[Series[q^(-1/3) eta[q^2]^2 eta[q^5]/eta[q], {q, 0, 50}], q] (* G. C. Greubel, Apr 17 2018 *)

Prog

(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^2*eta(x^5+A)/eta(x+A), n))}
(PARI) q='q+O('q^99); Vec(eta(q^2)^2*eta(q^5)/eta(q)) \\ Altug Alkan, Apr 18 2018

Crossrefs

Sequence in context: A089812 A260942 A260162 * A193261 A283497 A265507

Adjacent sequences: A123855 A123856 A123857 * A123859 A123860 A123861

Keyword

sign

Author

Michael Somos, Oct 13 2006

Status

approved