OFFSET
0,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q^(-1/2) * eta(q)^2 * eta(q^4)^9 / (eta(q^2)^5 * eta(q^8)^2) in powers of q.
Euler transform of period 8 sequence [ -2, 3, -2, -6, -2, 3, -2, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 512^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A113419.
a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(3^e) = (1 - (-3)^(e+1)) / 4, b(p^e) = (-1)^(e * [p/6]) * ((p*f)^(e+1) - 1) / (p*f - 1) where f = Kronecker( 18, p).
EXAMPLE
G.f. = 1 - 2*x + 4*x^2 - 8*x^3 + 7*x^4 - 10*x^5 + 12*x^6 - 8*x^7 + 18*x^8 + ...
G.f. = q - 2*q^3 + 4*q^5 - 8*q^7 + 7*q^9 - 10*q^11 + 12*q^13 - 8*q^15 + 18*q^17 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ q]^2 QPochhammer[ q^4]^9 / (QPochhammer[ q^2]^5 QPochhammer[ q^8]^2), {q, 0, n}]; (* Michael Somos, May 19 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^9 / (eta(x^2 + A)^5 * eta(x^8 + A)^2), n))};
(PARI) {a(n) = my(A, p, e, f); if( n<0, 0, A = factor(2*n + 1); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0^e, p==3, ((-p)^(e+1) - 1) / ((-p) - 1), p *= kronecker( 18, p); (-1)^(e*(p\6)) * (p^(e+1) - 1) / (p - 1))))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Mar 16 2012
STATUS
approved