OFFSET
-1,2
COMMENTS
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
FORMULA
Expansion of q^-1 * (phi(q) / psi(q^4))^2 in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of (eta(q^2)^5 / ( eta(q)^2* eta(q^4)* eta(q^8)^2 ))^2 in powers of q.
Euler transform of period 8 sequence [ 4, -6, 4, -4, 4, -6, 4, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u * (u-8) * (v-4) - (v-8)^2.
G.f.: (1/x)* Product_{k>0} (1 + x^k)^2 / ((1 + x^(2*k))^3 * (1 + x^(4*k))^2).
EXAMPLE
G.f. = 1/q + 4 + 4*q +2*q^3 - 8*q^5 - q^7 + 20*q^9 - 2*q^11 - 40*q^13 + 3*q^15 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ (2 EllipticTheta[ 3, 0, q] / EllipticTheta[ 2, 0, q^2])^2, {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)
eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q*(eta[q^2]^5/(eta[q]^2*eta[q^4]*eta[q^8]^2))^2, {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 17 2018 *)
PROG
(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^2 + A)^5 / (eta(x + A)^2 * eta(x^4 + A) * eta(x^8 + A)^2))^2, n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Jun 15 2007
STATUS
approved