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A123331 Expansion of (c(q)^2/(3c(q^2))-1)/2 in powers of q where c(q) is a cubic AGM function. 3
1, 2, 1, 1, 0, 2, 2, 2, 1, 0, 0, 1, 2, 4, 0, 1, 0, 2, 2, 0, 2, 0, 0, 2, 1, 4, 1, 2, 0, 0, 2, 2, 0, 0, 0, 1, 2, 4, 2, 0, 0, 4, 2, 0, 0, 0, 0, 1, 3, 2, 0, 2, 0, 2, 0, 4, 2, 0, 0, 0, 2, 4, 2, 1, 0, 0, 2, 0, 0, 0, 0, 2, 2, 4, 1, 2, 0, 4, 2, 0, 1, 0, 0, 2, 0, 4, 0, 0, 0, 0, 4, 0, 2, 0, 0, 2, 2, 6, 0, 1, 0, 0, 2, 4, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Table of n, a(n) for n=1..105.

FORMULA

Moebius transform is period 6 sequence [ 1, 1, 0, -1, -1, 0, ...].

a(n) is multiplicative with a(2^e) = (3-(-1)^e)/2, a(3^e) = 1, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6).

a(3n)=a(4n)=a(n). a(6n+5)=0.

G.f.: Sum_{k>0} x^k/(1-x^k+x^(2k)) = (theta_3(-q^3)^3/theta_3(-q) -1)/2.

PROG

(PARI) {a(n)=if(n<1, 0, -sumdiv(n, d, (-1)^d*kronecker(-3, d)))}

(PARI) {a(n)=local(A, p, e); if(n<1, 0, A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, (3-(-1)^e)/2, if(p==3, 1, if(p%6==1, e+1, !(e%2)))))))}

(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x^2+A)*eta(x^3+A)^6/ eta(x+A)^2/eta(x^6+A)^3-1)/2, n))}

CROSSREFS

Cf. A123330(n)=2*a(n) if n>0. A113974(n)=-(-1)^n*a(n).

Sequence in context: A122860 A113661 A113974 * A235141 A114638 A123340

Adjacent sequences:  A123328 A123329 A123330 * A123332 A123333 A123334

KEYWORD

nonn,mult

AUTHOR

Michael Somos, Sep 26 2006

STATUS

approved

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Last modified July 21 04:40 EDT 2019. Contains 325189 sequences. (Running on oeis4.)