OFFSET
0,2
COMMENTS
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 phi(-x) * phi(-x^2)^2 = phi(-x^2)^4 / phi(x) in powers of x where phi() is a Ramanujan theta function.
Expansion of eta(q^2)^3 * eta(q)^2 / eta(q^4)^2 in powers of q.
Euler transform of period 4 sequence [-2, -5, -2, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 32 (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A045828.
G.f.: Product_{k>0} (1 - x^(2*k))^3 * (1 - x^k)^2 / (1 - x^(4*k))^2.
EXAMPLE
G.f. = 1 - 2*q - 4*q^2 + 8*q^3 + 6*q^4 - 8*q^5 - 8*q^6 + 12*q^8 - 10*q^9 + ...
MATHEMATICA
a[n_]:= SeriesCoefficient[EllipticTheta[3, 0, q]* EllipticTheta[3, 0, -q]^2, {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Nov 30 2017 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x + A)^2 / eta(x^4 + A)^2, n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, May 29 2012
STATUS
approved