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Search: id:A132004
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| A132004 |
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Expansion of (1 -phi(q^3)/ phi(q) *phi(-q^2)* phi(-q^6))/2 in powers of q where phi() is a Ramanujan theta function. |
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+0
2
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| 1, -1, 1, -1, 2, -1, 0, -1, 1, -2, 0, -1, 2, 0, 2, -1, 2, -1, 0, -2, 0, 0, 0, -1, 3, -2, 1, 0, 2, -2, 0, -1, 0, -2, 0, -1, 2, 0, 2, -2, 2, 0, 0, 0, 2, 0, 0, -1, 1, -3, 2, -2, 2, -1, 0, 0, 0, -2, 0, -2, 2, 0, 0, -1, 4, 0, 0, -2, 0, 0, 0, -1, 2, -2, 3, 0, 0, -2, 0, -2, 1, -2, 0, 0, 4, 0, 2, 0, 2, -2, 0, 0, 0, 0, 0, -1, 2, -1, 0, -3, 2, -2, 0, -2, 0
(list; graph; listen)
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OFFSET
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1,5
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REFERENCES
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N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.72).
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FORMULA
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Expansion of (1 -eta(q)^2* eta(q^4)* eta(q^6)^7/( eta(q^2)^3* eta(q^3)^2* eta(q^12)^3))/2 in powers of q.
a(n) is multiplicative with b(2^e) = 2*0^e -1, b(3^e) = 1, b(p^e) = e+1 if p == 1 (mod 4), b(p^e) = (1+(-1)^e)/2 if p == 3 (mod 4).
a(3n)= a(n).
G.f.: Sum_{k>0} x^k/(x^k+1)* kronecker(-36, k).
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PROGRAM
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(PARI) {a(n)= if(n<1, 0, sumdiv(n, d, (-1)^(n+d)* kronecker(-36, d)))}
(PARI) {a(n)= local(A); if(n<0, 0, A= x*O(x^n); polcoeff( (1- eta(x+A)^2* eta(x^4+A)* eta(x^6+A)^7/( eta(x^2+A)^3* eta(x^3+A)^2* eta(x^12+A)^3))/2, n))}
(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==3, 1, if(p==2, -1, if(p%4==1, e+1, !(e%2)))))))}
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CROSSREFS
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A132003(n)= -2*a(n) unless n=0. A035154(n)= -a(2n). A125079(n)= a(2n+1).
Sequence in context: A035154 A113446 A121450 this_sequence A109294 A132966 A037897
Adjacent sequences: A132001 A132002 A132003 this_sequence A132005 A132006 A132007
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KEYWORD
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sign,mult
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AUTHOR
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Michael Somos, Aug 06 2007
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