0,13

Expansion of q^(-1)eta(q^3)eta(q^21) in powers of q^3.

Euler transform of period 7 sequence [ -1, -1, -1, -1, -1, -1, -2, ...]. - Michael Somos, Dec 06 2004

Given g.f. A(x), B(x)=x*A(x^3) satisfies 0=f(B(x), B(x^2), B(X^4)) where f(u, v, w)=u^2*w+2u*w^2-v^3 - Michael Somos Dec 06 2004

a(4n+1)=-a(n). - Michael Somos Dec 06 2004

G.f.: Product_{k>0} (1-x^k)(1-q^(7k)).

a(n)=b(3n+1) where b(n) is multiplicative and b(p^2e) = (-1)^e if p=2, b(p^e) = 0^e if p = 3, b(p^e) = (-1)^e if p = 7, b(p^e) = (1+(-1)^e)/2 if p == 3, 5, 6 (mod 7), else p == 1, 2, 4 (mod 7) and p=y^2+7x^2 when b(p^2e) = (-1)^e if x*y not divisible by 3, b(p^e) = e+1 if x divisible by 3 or (e+1)(-1)^e if y divisible by 3 . - Michael Somos May 28 2005

eta(q^3)eta(q^21) = q - q^4 - q^7 + q^16 + q^25 + q^28 - 2*q^37 ...

(PARI) a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff(eta(x+A)*eta(x^7+A), n))

(PARI) a(n)=if(n<0, 0, n=3*n+1; (qfrep([2, 1; 1, 32], n, 1)-qfrep([8, 1; 1, 8], n, 1))[n]) /* Michael Somos May 28 2005 */

(PARI) a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff(eta(x+A)*eta(x^7+A), n)) /* Michael Somos May 28 2005 */

(PARI) {a(n)=local(A, p, e, x, y); if(n<0, 0, n=3*n+1; A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, real(I^e), if(p==3, 0^e, if(p==7, (-1)^e, if(kronecker(p, 7)==-1, !(e%2), for(x=0, sqrtint(p\7), if(issquare(p-7*x^2, &y), y=if(x%3&y%3, real(I^e), (e+1)*if(x%3, (-1)^e, 1)); break)); y)))))))} /* Michael Somos May 28 2005 */

Sequence in context: A058087 A073274 A071957 this_sequence A064891 A035211 A035193

Adjacent sequences: A002652 A002653 A002654 this_sequence A002656 A002657 A002658

sign

N. J. A. Sloane (njas(AT)research.att.com).