OFFSET
0,2
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 eta(q)^2 * eta(q^8)^5 / (eta(q^2) * eta(q^4)^2 * eta(q^16)^2) in powers of q.
Euler transform of period 16 sequence [ -2, -1, -2, 1, -2, -1, -2, -4, -2, -1, -2, 1, -2, -1, -2, -2, ...].
Moebius transform is period 16 sequence [ -2, 2, 2, 4, -2, -2, 2, 0, -2, 2, 2, -4, -2, -2, 2, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 8 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A134013.
a(4*n + 2) = a(4*n + 3) = 0.
G.f.: 1 - 2 * ( x / (1 + x^2) + x^3 / (1 + x^6) - 2 * x^4 / (1 + x^8) + ... ).
EXAMPLE
G.f. = 1 - 2*q + 4*q^4 - 4*q^5 + 4*q^8 - 2*q^9 - 4*q^13 + 4*q^16 - 4*q^17 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0, q^4], {q, 0, n}]; (* Michael Somos, Oct 30 2015 *)
PROG
(PARI) {a(n) = if( n<1, n==0, if( n%4 < 2, (n%2*-6 + 4) * sumdiv(n, d, kronecker(-4, d))))};
(PARI) {a(n) = (-1)^n * if( n<1, n==0, 2 * qfrep([1, 0; 0, 4], n)[n])};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^8 + A)^5 / eta(x^2 + A) / eta(x^4 + A)^2 / eta(x^16 + A)^2, n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Oct 02 2007
STATUS
approved