OFFSET
1,3
COMMENTS
LINKS
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of eta(q) * eta(q^4) * eta(q^10) * eta(q^20) / (eta(q^2)^3 * eta(q^5)) in powers of q.
Euler transform of period 20 sequence [ -1, 2, -1, 1, 0, 2, -1, 1, -1, 2, -1, 1, -1, 2, 0, 1, -1, 2, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (80 t)) = 20^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A147701.
Convolution inverse of A145723.
a(n) ~ -(-1)^n * 3^(1/4) * exp(Pi*sqrt(3*n/5)) / (2^(5/2) * 5^(3/4) * n^(3/4)). - Vaclav Kotesovec, Jun 06 2018
EXAMPLE
G.f. = q - q^2 + 2*q^3 - 3*q^4 + 5*q^5 - 6*q^6 + 10*q^7 - 13*q^8 + 19*q^9 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ q QPochhammer[ q^20] QPochhammer[ -q^5, q^5] / QPochhammer[ -q], {q, 0, n}]; (* Michael Somos, Sep 05 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^10 + A) * eta(x^20 + A) / (eta(x^2 + A)^3 * eta(x^5 + A)), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Nov 10 2008
STATUS
approved