0,3

Expansion of f(q) * f(-q^2) = psi(-q) * phi(q) = chi(q) * f(-q^2)^2 = psi(q) * phi(-q^2) = f(q)^2 / chi(q) = f(q)^3 / phi(q) = f(-q^2)^3 / psi(-q) = phi(q)^2 / chi(q)^3 = chi(q)^3 * psi(-q)^2 = (f(q)^3 * psi(-q))^(1/2) = (f(-q^2)^3 * phi(q))^(1/2) in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.

Euler transform of period 4 sequence [ 1, -3, 1, -2, ...].

a(n) = b(8*n+1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = 0 if p == 3, 5, 7 (mod 8) and e odd, b(p^e) = 1 if p == 3 (mod 4) and e even, b(p^e) = (-1)^(e/2) if p == 5 (mod 8) and e even, b(p^e) = e+1 if p == 1 (mod 8) and p = x^2 + 64*y^2, b(p^e) = (-1)^e * (e+1) if p == 1 (mod 8) and p is not of the form x^2 + 64*y^2.

a(9*n+1) = a(n), a(9*n+4) = a(9*n+7) = 0.

G.f.: Product_{k>0} (1 - x^(2*k))^2 * (1 + x^(2*k-1)).

q + q^9 - 2*q^17 - q^25 - 2*q^41 + q^49 + 2*q^73 + q^81 + 2*q^89 - 2*q^97 + ...

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 / eta(x + A) / eta(x^4 + A), n))}

(PARI) {a(n) = local(A, p, e); if( n<0, 0, n = 8*n + 1; A = factor(n); prod(k=1, matsize(A)[1], if(p = A[k, 1], e = A[k, 2]; if( p==2, 0, if( p%8==1, (e+1) * if( qfbclassno( -8 * p) / 4 % 2, (-1)^e, 1), if( e%2==0, (-1)^(e/2 * (p%8==5)))))))) }

(PARI) {a(n) = if( n<0, 0, n = 8*n + 1; (qfrep([1, 0; 0, 64], n) - qfrep([4, 2; 2, 17], n))[n])}

(-1)^n * A030204(n) = a(n).

Sequence in context: A156319 A083650 A030204 this_sequence A143540 A030200 A095734

Adjacent sequences: A138511 A138512 A138513 this_sequence A138515 A138516 A138517

sign

Michael Somos, Mar 22 2008