A121362 Expansion of eta(q)*eta(q^6)*eta(q^10)*eta(q^15)/(eta(q^3)*eta(q^5)) in powers of q.

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

Offset: 1

Author

Michael Somos, Jul 22 2006

Keywords

• sign
• mult

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 = 1..1000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A082451(n) = |a(n)|.

Programs

Mathematica

eta[q_] := q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[eta[q]* eta[q^6]*eta[q^10]*eta[q^15]/(eta[q^3]*eta[q^5]), {q, 0, n}]; Table[a[n], {n,1,50}] (* G. C. Greubel, Feb 11 2018 *)

PARI

{a(n)=local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff( eta(x+A)*eta(x^6+A)*eta(x^10+A)*eta(x^15+A)/(eta(x^3+A)*eta(x^5+A)), n))}

Formula

Euler transform of period 30 sequence [ -1, -1, 0, -1, 0, -1, -1, -1, 0, -1, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, 0, -1, -1, -1, 0, -1, 0, -1, -1, -2, ...].

Expansion of q*f(-q)f(-q^15)/(chi(-q^3)chi(-q^5)) in powers of q where f(),chi() are Ramanujan theta functions.

G.f.: x Product_{n>0} (1-x^n)(1+x^(3n))(1+x^(5n))(1-x^(15n)).

a(n) is multiplicative with a(2^e)=a(3^e)=a(5^e)=(-1)^e, a(p^e) = e+1 if p == 1,4 (mod 15), a(p^e) = (-1)^e*(e+1) if p == 2,8 (mod 15), a(p^e) = (1+( -1)^e)/2 if p == 7,11,13,14 (mod 15).