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A134015
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Expansion of (1 - phi(-q) * phi(q^4)) / 2 in powers of q where phi() is a Ramanujan theta function.
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3
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1, 0, 0, -2, 2, 0, 0, -2, 1, 0, 0, 0, 2, 0, 0, -2, 2, 0, 0, -4, 0, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, -2, 0, 0, 0, -2, 2, 0, 0, -4, 2, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, -4, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, -2, 4, 0, 0, -4, 0, 0, 0, -2, 2, 0, 0, 0, 0, 0, 0, -4, 1, 0, 0
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OFFSET
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1,4
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COMMENTS
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LINKS
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FORMULA
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Moebius transform is period 16 sequence [ 1, -1, -1, -2, 1, 1, -1, 0, 1, -1, -1, 2, 1, 1, -1, 0, ...].
a(n) is multiplicative with a(2) = 0, a(2^e) = -2 if e>1, a(p^e) = e+1 if p == 1 (mod 4), a(p^e) = (1+(-1)^e)/2 if p == 3 (mod 4).
a(4*n+2) = a(4*n+3) = 0.
G.f.: x / (1 + x^2) + x^3 / (1 + x^6) - 2 * x^4 / (1 + x^8) + ...
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EXAMPLE
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G.f. = x - 2*x^4 + 2*x^5 - 2*x^8 + x^9 + 2*x^13 - 2*x^16 + 2*x^17 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ (1 - EllipticTheta[ 4, 0, x] EllipticTheta[ 3, 0, x^4]) / 2, {x, 0, n}]; (* Michael Somos, Oct 28 2015 *)
a[ n_] := If[ n < 1 || Mod[n, 4] > 1, 0, (Mod[n, 2] 3 - 2) DivisorSum[ n, KroneckerSymbol[ -4, #]&]]; (* Michael Somos, Oct 28 2015 *)
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PROG
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(PARI) {a(n) = if( n<1 || n%4>1, 0, (n%2*3 - 2) * sumdiv(n, d, kronecker(-4, d)))};
(PARI) {a(n) = -(-1)^n * if( n<1, 0, qfrep([1, 0; 0, 4], n)[n])};
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CROSSREFS
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KEYWORD
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sign,mult
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AUTHOR
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STATUS
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approved
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