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A227395
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Expansion of q^2 * phi(-q) * psi(q^16) in powers of q where phi(), psi() are Ramanujan theta functions.
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4
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1, -2, 0, 0, 2, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 3, -2, 0, 0, 2, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 1, -4, 0, 0, 4, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 4, -2, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 2, -2, 0
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OFFSET
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2,2
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COMMENTS
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LINKS
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FORMULA
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Expansion of eta(q)^2 * eta(q^32)^2 / (eta(q^2) * eta(q^16)) in powers of q.
Euler transform of period 32 sequence [ -2, -1, -2, -1, -2, -1, -2, -1, -2, -1, -2, -1, -2, -1, -2, 0, -2, -1, -2, -1, -2, -1, -2, -1, -2, -1, -2, -1, -2, -1, -2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 32^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
a(4*n) = a(4*n + 1) = a(8*n + 7) = 0. a(4*n + 2) = A113411(n). a(8*n + 3) = -2 * A033761(n).
G.f.: x^2 * Product_{k>0} (1 - x^k)^2 * (1 - x^(32*k))^2 / ((1 - x^(2*k)) * (1 - x^(16*k))).
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EXAMPLE
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G.f. = q^2 - 2*q^3 + 2*q^6 - 2*q^11 + 3*q^18 - 2*q^19 + 2*q^22 - 4*q^27 + 2*q^34 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 2, 0, q^8] / 2, {q, 0, n}];
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PROG
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(PARI) {a(n) = local(A); if( n<2, 0, n -= 2; A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^32 + A)^2 / (eta(x^2 + A) * eta(x^16 + A)), 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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