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 f(x^2)^2 * chi(-x) in powers of x where chi(), f() are Ramanujan theta functions.
Expansion of q^(-1/8) * eta(q) * eta(q^4)^6 / (eta(q^2)^3 * eta(q^8)^2) in powers of q.
Euler transform of period 8 sequence [ -1, 2, -1, -4, -1, 2, -1, -2, ...].
a(n) = b(8*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(p^e) = (-1)^(e * (p+1)/8) * (1 + (-1)^e) / 2 if p == 3 (mod 4), b(p^e) = (e+1) * I^(e * (p-1)/4) if p == 1 (mod 4).
a(9*n + 4) = a(9*n + 7) = 0. a(9*n + 1) = -a(n).
a(n) = (-1)^n * A113407(n).
EXAMPLE
1 - x + 2*x^2 - 3*x^3 - 2*x^5 + x^6 + 4*x^8 - 2*x^9 + x^10 - 2*x^11 + ...
q - q^9 + 2*q^17 - 3*q^25 - 2*q^41 + q^49 + 4*q^65 - 2*q^73 + q^81 - 2*q^89 + ...
MATHEMATICA
a[n_]:= SeriesCoefficient[QPochhammer[-q^2, -q^2]^2*QPochhammer[q, q^2], {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Dec 07 2017 *)
PROG
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A)^6 / (eta(x^2 + A)^3 * eta(x^8 + A)^2), n))}
(PARI) {a(n) = local(A, p, e); if( n<0, 0, n = 8*n + 1; A = factor(n); simplify( prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 0, if( p%4 == 3, if( e%2, 0, (-1)^(e * (p+1) / 8)), (e+1) * I^(e * (p-1)/ 4)))))))}
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, May 30 2013
STATUS
approved