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A097109
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G.f.: s(2)^2*s(3)^3/(s(1)*s(6)^2), where s(k) := subs(q=q^k, eta(q)) and eta(q) is Dedekind's function, cf. A010815.
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2
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1, 1, 0, -2, -3, 0, 0, 2, 0, -2, 0, 0, 6, 2, 0, 0, -3, 0, 0, 2, 0, -4, 0, 0, 0, 1, 0, -2, -6, 0, 0, 2, 0, 0, 0, 0, 6, 2, 0, -4, 0, 0, 0, 2, 0, 0, 0, 0, 6, 3, 0, 0, -6, 0, 0, 0, 0, -4, 0, 0, 0, 2, 0, -4, -3, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, -2, -6, 0, 0, 2, 0, -2, 0, 0, 12, 0, 0, 0, 0, 0, 0, 4, 0, -4, 0, 0, 0, 2, 0, 0, -3, 0, 0, 2, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Coefficients are multiplicative [Fine].
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REFERENCES
| N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 80, Eq. (32.36).
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FORMULA
| Fine gives an explicit formula for a(n) in terms of the divisors of n.
Expansion of (a(q)-3a(q^3)-4a(q^4)+12a(q^12))/6 in powers of q where a() is a cubic AGM analog function. - Michael Somos Sep 15 2006
Euler transform of period 6 sequence [ 1, -1, -2, -1, 1, -2, ...]. - Michael Somos Sep 15 2006
a(n) is multiplicative with a(2^e) = -3(1+(-1)^e)/2 if e>0, 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) - Michael Somos Sep 15 2006
a(3n+2)=0.
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PROG
| (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^2*eta(x^3+A)^3/eta(x+A)/eta(x^6+A)^2, n))} /* Michael Somos Sep 15 2006 */
(PARI) {a(n)= local(A, p, e); if(n<1, n==0, A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, 3*(e%2-1), if(p==3, -2, if(p%6==1, e+1, 1-e%2))))))} /* Michael Somos Sep 15 2006 */
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CROSSREFS
| A115978(n)=a(3n). A122861(n)=a(3n+1).
Sequence in context: A069585 A167613 A078332 * A160018 A175099 A151867
Adjacent sequences: A097106 A097107 A097108 * A097110 A097111 A097112
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KEYWORD
| sign,mult
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Sep 16 2004
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