A123758 Expansion of q^(-1/3)*eta(q)*eta(q^4)*eta(q^5)/eta(q^2) in powers of q.

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

Offset

0,7

Comments

Ramanujan theta functions: f(q) := Product_{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) := Product_{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 20 sequence [ -1, 0, -1, -1, -2, 0, -1, -1, -1, -1, -1, -1, -1, 0, -2, -1, -1, 0, -1, -2, ...].

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

a(8n+2) = a(8n+4) = 0.

Mathematica

eta[q_] := q^(1/24)*QPochhammer[q]; CoefficientList[Series[q^(-1/3) eta[q] eta[q^4] eta[q^5]/eta[q^2], {q, 0, 100}], q] (* G. C. Greubel, Apr 19 2018 *)

Prog

(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)*eta(x^4+A)*eta(x^5+A)/eta(x^2+A), n))}
(PARI) {a(n) = local(s, k); if(n<0, 0, n=24*n+8; for(j=1, sqrtint(n\5), if((j^2%6==1)& issquare( (n-5*j^2)/3, &k)& (k%2), s+= (-1)^((j+1)\6+ (k+2)\4))); s)}

Crossrefs

Sequence in context: A070103 A113048 A331671 * A231642 A288318 A354099

Adjacent sequences: A123755 A123756 A123757 * A123759 A123760 A123761

Keyword

sign

Author

Michael Somos, Oct 12 2006

Status

approved