OFFSET
0,3
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q^(-11/24) * eta(q^8)^2 / (eta(q) * eta(q^4)) in powers of q.
Euler transform of period 8 sequence [1, 1, 1, 2, 1, 1, 1, 0, ...].
G.f.: (Sum_{k>=1} x^(2*k*(k-1))) / (Product_{k>=1} (1 - x^k)).
a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(13/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Nov 16 2017
Expansion of (chi(q)^2 - chi(-q)^2)/(4*q) in powers of q^2 where chi() is a Ramanujan theta function. - Michael Somos, Nov 02 2019
EXAMPLE
G.f. = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 8*x^5 + 13*x^6 + 18*x^7 + 27*x^8 + 37*x^9 + ...
G.f. = q^11 + q^35 + 2*q^59 + 3*q^83 + 6*q^107 + 8*q^131 + 13*q^155 + 18*q^179 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q^2] / (2 q^(1/2) QPochhammer[ q]), {q, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^8 + A)^2 / (eta(x + A) * eta(x^4 + A)), n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Aug 31 2013
STATUS
approved