OFFSET
-1,9
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 * chi(-q) * chi(-q^23) in powers of q where chi() is a Ramanujan theta function.
Expansion of eta(q) * eta(q^23) / (eta(q^2) * eta(q^46)) in powers of q.
Euler transform of period 46 sequence [ -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -2, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 * v - v^2 + 2 * u + 2 * u * v.
G.f. is a period 1 Fourier series which satisfies f(-1 / (46 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A092833.
G.f.: x^-1 * (Product_{k>0} (1 + x^k) * (1 + x^(23*k)))^-1.
a(n) = A058688(n) unless n = 0.
Convolution inverse of A092833. - Michael Somos, Mar 23 2015
a(n) ~ -(-1)^n * exp(2*Pi*sqrt(n/23)) / (2 * 23^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jun 06 2018
EXAMPLE
G.f. = 1/q - 1 - q^2 + q^3 - q^4 + q^5 - q^6 + 2*q^7 - 2*q^8 + 2*q^9 - 2*q^10 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ q, q^2] QPochhammer[ q^23, q^46] / q, {q, 0, n}]; (* Michael Somos, Mar 23 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^23 + A) / (eta(x^2 + A) * eta(x^46 + A)), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Aug 18 2007
STATUS
approved