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A110399
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Expansion of (theta_3(q)theta_3(q^7)-1)/2 in powers of q.
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0
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1, 0, 0, 1, 0, 0, 1, 2, 1, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 1, 2, 0, 0, 4, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 5, 0, 0, 2, 0, 0, 0, 2, 2, 0, 0, 0, 0, 2, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0
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OFFSET
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1,8
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REFERENCES
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B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 302 Entry 17(ii)
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LINKS
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Table of n, a(n) for n=1..105.
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FORMULA
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a(n) is multiplicative and a(2^e) = |e-1|, a(7^e)= 1, a(p^e) = e+1 if p == 1, 2, 4 (mod 7), a(p^e) = (1+(-1)^e)/2 if p == 3, 5, 6 (mod 7).
G.f.: Sum_{k>0}, kronecker(-7, k) x^k/(1-(-x)^k).
G.f.: (theta_3(q)theta_3(q^7)-1)/2 where theta_3(q)=1+2(q+q^4+q^9+...).
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PROG
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(PARI) {a(n)=local(x); if(n<1, 0, x=valuation(n, 2); abs(x-1)*sumdiv(n/2^x, d, kronecker(-28, 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, e-1, if(p==7, 1, if(kronecker(-7, p)==-1, (1+(-1)^e)/2, e+1))))))}
(PARI) {a(n)=local(A); if(n<1, 0, A=x*O(x^n); polcoeff( (eta(x+A)^-2*eta(x^2+A)^5 *eta(x^4+A)^-2*eta(x^7+A)^-2*eta(x^14+A)^5 *eta(x^28+A)^-2-1)/2, n))}
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CROSSREFS
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A033719(n)=2 a(n) if n>0. A035162(2n+1)=A035182(2n+1)=a(2n+1).
Sequence in context: A058998 A085324 A062154 * A193275 A182033 A112214
Adjacent sequences: A110396 A110397 A110398 * A110400 A110401 A110402
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KEYWORD
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nonn,mult
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AUTHOR
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Michael Somos, Oct 22 2005
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STATUS
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approved
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