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A143434
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Expansion of f(x, -x^3) in powers of x where f(,) is Ramanujan's two-variable theta function.
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2
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1, 1, 0, -1, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
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OFFSET
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0,1
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LINKS
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FORMULA
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Euler transform of period 16 sequence [ 1, -1, -1, 1, -1, 0, 1, -2, 1, 0, -1, 1, -1, -1, 1, -1, ...].
G.f.: Sum_{k>=0} (-1)^floor((k + 2) / 4) * x^(k * (k+1) / 2).
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EXAMPLE
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1 + x - x^3 - x^6 - x^10 - x^15 + x^21 + x^28 + x^36 + x^45 - x^55 - x^66 + ...
q + q^9 - q^25 - q^49 - q^81 - q^121 + q^169 + q^225 + q^289 + q^361 + ...
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MATHEMATICA
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a[ n_] := If[ n < 0, 0, SeriesCoefficient[ (Series[ EllipticTheta[ 3, Log[y] / (2 I), I x^2], {x, 0, n + Floor@Sqrt[n]}] // Normal // TrigToExp) /. {y -> I x}, {x, 0, n}]]
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PROG
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(PARI) {a(n) = if( n<0, 0, if( issquare( 8*n + 1, &n), n = n\2; (-1)^((n + 2) \ 4), 0))}
(PARI) {a(n) = local(A); if( n<0, 0, polcoeff( prod( k=1, n, (1 - x^k)^( [1, -1, 1, 1, -1, 1, 0, -1, 2, -1, 0, 1, -1, 1, 1, -1] [k%16 + 1]), 1 + x * O(x^n)), n))}
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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