OFFSET
1,2
COMMENTS
Number 12 of the 14 eta-quotients listed in Table 2 of Moy 2013. - Michael Somos, Sep 19 2013
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..1000
Kevin Acres, David Broadhurst, Eta quotients and Rademacher sums, arXiv:1810.07478 [math.NT], 2018. See Table 1 p. 10.
Richard Moy, Congruences among power series coefficients of modular forms, arXiv:1309.4320
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of eta(q^2) * eta(q^16)^2 / (eta(q)^2 * eta(q^8)) in powers of q.
Euler transform of period 16 sequence [ 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v * (1 + 4*u) * (1 + 2*v).
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 1/8 * g(t) where q = exp(2 Pi i t) and g() is g.f. for A185338.
a(n) is odd iff n is an odd square. If n>2 is a power of 2 then the highest power of 2 dividing a(n) is (n/2)^3. - Michael Somos, Feb 18 2007
G.f.: x * Product_{k>0} (1 + x^k)^2 * (1 + x^(2*k)) * (1 + x^(4*k)) * (1 + x^(8*k))^2. Michael Somos, Sep 19 2013
a(n) ~ exp(sqrt(n)*Pi) / (2^(9/2) * n^(3/4)). - Vaclav Kotesovec, Nov 15 2017
EXAMPLE
G.f. = q + 2*q^2 + 4*q^3 + 8*q^4 + 14*q^5 + 24*q^6 + 40*q^7 + 64*q^8 + 101*q^9 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q^4] / EllipticTheta[ 4, 0, q] / 2, {q, 0, n}]; (* Michael Somos, Sep 19 2013 *)
PROG
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^16 + A)^2 / (eta(x + A)^2 * eta(x^8 + A)), n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Oct 04 2006
STATUS
approved