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A112848
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Expansion of eta(q)*eta(q^2)*eta(q^18)^2/(eta(q^6)*eta(q^9)) in powers of q.
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1
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1, -1, -2, 1, 0, 2, 2, -1, -2, 0, 0, -2, 2, -2, 0, 1, 0, 2, 2, 0, -4, 0, 0, 2, 1, -2, -2, 2, 0, 0, 2, -1, 0, 0, 0, -2, 2, -2, -4, 0, 0, 4, 2, 0, 0, 0, 0, -2, 3, -1, 0, 2, 0, 2, 0, -2, -4, 0, 0, 0, 2, -2, -4, 1, 0, 0, 2, 0, 0, 0, 0, 2, 2, -2, -2, 2, 0, 4, 2, 0, -2, 0, 0, -4, 0, -2, 0, 0, 0, 0, 4, 0, -4, 0, 0, 2, 2, -3, 0, 1, 0, 0, 2, -2, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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FORMULA
| Euler transform of period 18 sequence [ -1, -2, -1, -2, -1, -1, -1, -2, 0, -2, -1, -1, -1, -2, -1, -2, -1, -2, ...].
Moebius transform is period 18 sequence [1, -2, -3, 2, -1, 6, 1, -2, 0, 2, -1, -6, 1, -2, 3, 2, -1, 0, ...].
Multiplicative with a(2^e) = (-1)^e, a(3^e) = -2 if e>0, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6).
G.f.: Sum_{k>0} kronecker(-3, k) x^k(1-x^(2k))^2/(1-x^(6k)) = x Product_{k>0} (1-x^k)(1-x^(2k))(1+x^(9k))(1+x^(6k)+x^(12k)).
a(3n)=-2*A092829(n). a(3n+1)=A093829(3n+1)=A033687(n). a(3n+2)=A093829(3n+2). a(6n)/2=A093829(n). a(6n+1)=A097195(n). a(6n+3)=-2*A033762(n). a(6n+5)=0.
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PROG
| (PARI) {a(n)=if(n<1, 0, if(n%3==0, n/=3; -2, 1)* sumdiv(n, d, kronecker(-12, d) -if(d%2==0, 2*kronecker(-3, d/2))))}
(PARI) {a(n)=local(A); if (n<1, 0, n--; A=x*O(x^n); polcoeff( eta(x+A)*eta(x^2+A)*eta(x^18+A)^2/ eta(x^6+A)/eta(x^9+A), n))}
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CROSSREFS
| Sequence in context: A113423 A131258 A029366 * A198727 A035152 A035204
Adjacent sequences: A112845 A112846 A112847 * A112849 A112850 A112851
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KEYWORD
| sign,mult
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AUTHOR
| Michael Somos, Sep 22 2005
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