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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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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
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OFFSET
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1,5
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COMMENTS
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Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
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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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LINKS
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Table of n, a(n) for n=1..105.
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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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).
G.f.: Sum_{k>0} x^k / (1 + x^k) * kronecker( -36, k).
a(3*n) = a(n). -2 * a(n) = A132003(n) unless n=0. -a(2*n) = A035154(n). a(2*n + 1) = A125079(n).
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EXAMPLE
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x - x^2 + x^3 - x^4 + 2*x^5 - x^6 - x^8 + x^9 - 2*x^10 - x^12 + 2*x^13 + ...
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PROG
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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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Cf. A035154, A125079, A132003.
Sequence in context: A035154 A113446 A121450 * A143110 A109294 A132966
Adjacent sequences: A132001 A132002 A132003 * 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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STATUS
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approved
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