OFFSET
0,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Hossein Movasati, Younes Nikdelan, Product formulas for weight two newforms, arXiv:1803.01414 [math.NT], 2018.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q^(-1/2) * ((eta(q^2)^5 / eta(q^4))^2 + 8 * (eta(q^4)^5 / eta(q^2))^2) in powers of q.
Expansion of q^(-1/2) * (eta(q^2)^12 + 8 * eta(q^4)^12) / ( eta(q^2) * eta(q^4) )^2 in powers of q.
a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(p^e) = b(p) * b(p^(e-1)) - p^3 * b(p^(e-2)) if p>2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 8^2 (t / i)^4 f(t) where q = exp(2 Pi i t).
If F(x) is the g.f. for A002171, then A(x) * F(x^2) = B(x) the g.f. for A227239. - Michael Somos, Jan 08 2015
EXAMPLE
G.f. = 1 + 8*x - 10*x^2 + 16*x^3 + 37*x^4 - 40*x^5 - 50*x^6 - 80*x^7 - 30*x^8 + ...
G.f. = q + 8*q^3 - 10*q^5 + 16*q^7 + 37*q^9 - 40*q^11 - 50*q^13 - 80*q^15 - 30*q^17 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2]^12 + 8 x QPochhammer[ x^4]^12) / (QPochhammer[ x^2] QPochhammer[ x^4])^2, {x, 0, n}];
PROG
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^5 / eta(x^4 + A))^2 + 8 * x * (eta(x^4 + A)^5 / eta(x^2 + A))^2, n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Sep 02 2013
STATUS
approved