OFFSET
0,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of (eta(q) * eta(q^16) * eta(q^4)^7)^2 / (eta(q^2 ) * eta(q^8))^9 in powers of q.
Euler transform of period 16 sequence [-2, 7, -2, -7, -2, 7, -2, 2, -2, 7, -2, -7, -2, 7, -2, 0, ...].
G.f.: Product_{k>0} (1 + x^(4*k - 2))^5 / ((1 + x^(2*k - 1)) * (1 + x^(8*k - 4)))^2.
a(n) ~ (-1)^n * exp(sqrt(n)*Pi)/(4*sqrt(2)*n^(3/4)). - Vaclav Kotesovec, Sep 08 2017
EXAMPLE
G.f. = 1 - 2*q + 8*q^2 - 16*q^3 + 32*q^4 - 60*q^5 + 96*q^6 - 160*q^7 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^2]^2 / (EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^4]), {q, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ -q^2, q^4]^5 / (QPochhammer[ -q, q^2] QPochhammer[ -q^4, q^8])^2, {q, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^16 + A) * eta(x^4 + A)^7)^2 / (eta(x^2 + A) * eta(x^8 + A))^9, n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Nov 23 2013
STATUS
approved