login
A249371
Expansion of q^3 * f(q) * f(-q^4) * f(q^15) * f(-q^60) * chi(-q^3) * chi(-q^5) in powers of q where chi(), f() are Ramanujan theta functions.
2
1, 1, -1, -1, -2, -2, 0, 1, 2, 0, 2, 2, 0, 1, 2, 0, -4, 0, -2, -2, 0, 1, 0, -2, 1, -2, -2, 0, 4, 2, -2, -2, 2, 1, 6, 4, 0, -1, -8, 2, -4, -2, -1, 0, -4, 0, 8, 0, 4, 0, 2, -1, 2, 2, 0, 2, 2, -1, -4, -4, -2, -3, 0, 2, -8, 4, 0, -2, 4, -2, -4, -6, 1, 0, 0, 0, 4
OFFSET
3,5
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
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion (1/3) * b(q^2) * c(q^10) * (c(q) * b(q^5) / (b(q) * c(q^5)))^(1/4) in powers of q where b(), c() are cubic AGM theta functions.
Expansion of eta(q^2)^3 * eta(q^3) * eta(q^5) * eta(q^30)^3 / (eta(q) * eta(q^6) * eta(q^10) * eta(q^15)) in powers of q.
Euler transform of period 30 sequence [ 1, -2, 0, -2, 0, -2, 1, -2, 0, -2, 1, -2, 1, -2, 0, -2, 1, -2, 1, -2, 0, -2, 1, -2, 0, -2, 0, -2, 1, -4, ...].
EXAMPLE
G.f. = q^3 + q^4 - q^5 - q^6 - 2*q^7 - 2*q^8 + q^10 + 2*q^11 + 2*q^13 + ...
MATHEMATICA
eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[ eta[q^2]^3* eta[q^3]*eta[q^5]*eta[q^30]^3/(eta[q]*eta[q^6]*eta[q^10]*eta[q^15]), {q, 0, 60}], q]] (* G. C. Greubel, Aug 13 2018 *)
PROG
(PARI) {a(n) = my(A); n-=3; if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^3 + A) * eta(x^5 + A) * eta(x^30 + A)^3 / (eta(x + A) * eta(x^6 + A) * eta(x^10 + A) * eta(x^15 + A)), n))};
(Magma) Basis( CuspForms( Gamma0(30), 2), 89) [3];
CROSSREFS
Sequence in context: A182882 A134178 A059018 * A122190 A237291 A146093
KEYWORD
sign
AUTHOR
Michael Somos, Oct 26 2014
STATUS
approved