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A226860
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Expansion of psi(-x) * phi(-x^6) in powers of x where phi(), psi() are Ramanujan theta functions.
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2
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1, -1, 0, -1, 0, 0, -1, 2, 0, 2, 1, 0, -2, 0, 0, -1, -2, 0, 0, 0, 0, 1, 0, 0, 2, -2, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, -2, 0, 0, -2, 0, 0, -3, 0, 0, 0, 0, 0, 2, 2, 0, -2, 1, 0, 2, 0, 0, 0, 2, 0, 0, -2, 0, 1, 0, 0, 0, 0, 0, -2, 0, 0, 2, 0, 0, 1, -2, 0, 0
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OFFSET
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0,8
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COMMENTS
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LINKS
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FORMULA
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Expansion of q^(-1/8) * eta(q) * eta(q^4) * eta(q^6)^2 / (eta(q^2) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [-1, 0, -1, -1, -1, -2, -1, -1, -1, 0, -1, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (192 t)) = 96^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A226350.
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EXAMPLE
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G.f. = 1 - x - x^3 - x^6 + 2*x^7 + 2*x^9 + x^10 - 2*x^12 - x^15 - 2*x^16 + x^21 + ...
G.f. = q - q^9 - q^25 - q^49 + 2*q^57 + 2*q^73 + q^81 - 2*q^97 - q^121 - 2*q^129 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^6] EllipticTheta[ 2, Pi/4, q^(1/2)] / (Sqrt[2] q^(1/8)), {q, 0, n}];
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PROG
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(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^2 / (eta(x^2 + A) * eta(x^12 + A)), n))};
(PARI) {a(n) = my(A, p, e, i); if( n<0, 0, n = 8*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0, p==3, I^e, p%24 == 1 || p%24==19, for(j=1, sqrtint(p\18), if( issquare( p - 18*j^2, &i), break)); (e+1) * (if(p%24==1, 1, -I) * kronecker( 12, i))^e, if( e%2, 0, if(p%24>12, 1, -1)^(e/2))) ))}; /* Michael Somos, Sep 08 2014 */
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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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