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A129134
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Expansion of (1 - phi(-q) * phi(-q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.
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1
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1, 1, -2, -1, 0, 2, 0, -1, 3, 0, -2, -2, 0, 0, 0, -1, 2, 3, -2, 0, 0, 2, 0, -2, 1, 0, -4, 0, 0, 0, 0, -1, 4, 2, 0, -3, 0, 2, 0, 0, 2, 0, -2, -2, 0, 0, 0, -2, 1, 1, -4, 0, 0, 4, 0, 0, 4, 0, -2, 0, 0, 0, 0, -1, 0, 4, -2, -2, 0, 0, 0, -3, 2, 0, -2, -2, 0, 0, 0, 0, 5, 2, -2, 0, 0, 2, 0, -2, 2, 0, 0, 0, 0, 0, 0, -2, 2, 1, -6, -1, 0, 4, 0, 0, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
For n nonzero, a(n) is nonzero if and only if n is in A002479.
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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| Expansion of (1 - eta(q)^2 * eta(q^2) / eta(q^4)) / 2 in powers of q.
G.f.: (1 - Product_{k>0} (1 - x^k)^2 / (1 + x^(2*k)) )/2.
a(n) = A002325(n) * (-1)^[(n-1)/2]. A082564(n) = -2 * a(n) unless n=0.
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EXAMPLE
| q + q^2 - 2*q^3 - q^4 + 2*q^6 - q^8 + 3*q^9 - 2*q^11 - 2*q^12 - q^16 + ...
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PROG
| (PARI) {a(n) = if( n<1, 0, (-1)^((n-1)\2) * sumdiv(n, d, kronecker( -8, d)))}
(PARI) {a(n) = local(A); if( n<1, 0, A = x * O(x^n); polcoeff( (1 - eta(x + A)^2 * eta(x^2 + A) / eta(x^4 + A)) / 2, n))}
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CROSSREFS
| Cf. A002325, A002479, A082564.
Sequence in context: A036577 A002325 * A133693 A065675 A194313 A127476
Adjacent sequences: A129131 A129132 A129133 * A129135 A129136 A129137
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Mar 30 2007
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