OFFSET
0,3
COMMENTS
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
O-Yeat Chan, Some asymptotics for cranks, Acta Arithmetica 120 (2005), p. 107-143.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of phi(x) * chi(-x) = phi(-x^2) * chi(x) = psi(x) * chi(-x^2)^2 = f(-x) * chi(x)^2 = f(x) * chi(-x^2) = f(x)^2 / psi(x) = phi(-x^2)^2 / f(-x) if powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.
Expansion of q^(1/24) * eta(q^2)^4 / (eta(q^4)^2 * eta(q)) in powers of q.
Euler transform of period 4 sequence [ 1, -3, 1, -1, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = 96^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A096920.
G.f.: Product_{k>0} (1 - x^k) * (1 + x^(2*k-1))^2.
a(n) ~ c1(n) * exp(Pi*sqrt(2*(n - 1/24)/3)/4) / (2*sqrt(2*(n - 1/24))), where c1(n) = (-1)^(n/2) * 1.847759... if n is even and c1(n) = -(-1)^((n+1)/2) * 0.765366... if n is odd [Chan, 2005, formula 2.2 and 2.7, where a1(n) = a(n) if n is even and a1(n) = -a(n) if n is odd]. - Vaclav Kotesovec, May 08 2020
In closed form, abs(c1(n)) = sqrt(2 + (-1)^n*sqrt(2)). - Vaclav Kotesovec, May 08 2020
EXAMPLE
G.f. = 1 + x - 2*x^2 - x^3 + x^4 - x^5 x^6 - x^7 + 2*x^8 + 2*x^9 - 2*x^10 + ...
G.f. = 1/q + q^23 - 2*q^47 - q^71 + q^95 - q^119 - q^143 - q^167 + 2*q^191 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ x, x^2], {x, 0, n}]; (* Michael Somos, Aug 31 2014 *)
CoefficientList[Series[QPochhammer[-x] * QPochhammer[x^2] / QPochhammer[x^4], {x, 0, 100}], x] (* Vaclav Kotesovec, May 08 2020 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 / (eta(x + A) * eta(x^4 + A)^2), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Mar 24 2008
STATUS
approved