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A254613
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Expansion of f(-x^3)^4 / (f(-x) * f(-x^9)) in powers of x where f() is a Ramanujan theta function.
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2
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1, 1, 2, -1, 1, -1, 1, -3, -2, 1, 1, 0, -3, -1, 0, -2, -1, 0, 0, 1, 1, 0, 0, -2, 2, 0, -1, 2, -2, 2, 1, 2, 0, -1, 1, 1, -1, 3, 1, 0, -1, 3, 0, 0, 1, 1, 0, -3, -1, 0, -1, 2, 1, 1, 0, -1, -1, 3, 0, 1, 0, -2, -3, -2, -1, 0, -1, 0, 0, -2, 2, 2, -3, -1, 0, 0, 0
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OFFSET
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0,3
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COMMENTS
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LINKS
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FORMULA
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Expansion of q^(-1/12) * eta(q^3)^4 / (eta(q) * eta(q^9)) in powers of q.
Euler transform of period 9 sequence [1, 1, -3, 1, 1, -3, 1, 1, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (1296 t)) = 36 (t/i) f(t) where q = exp(2 Pi i t).
a(n) = b(12*n + 1) and A254612(n) = b(12*n + 5) / sqrt(3) where b() is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 if p == 7,11 (mod 12), b(p^e) = b(p)*b(p^(e-1)) - b(p^(e-2)) if p == 1,5 (mod 12) where b(p) = sqrt(3) * k12(x,y) * (if 3|y then 0 else 1) with 2*p = x^2 + 9*y^2 if p == 5 (mod 12) and b(p) = k12(9*y+x, y-x) + k12(9*y-x, y+x) with p = x^2 + 9*y^2 if p == 1 (mod 12) where k12(x,y) := Kronecker(12, x) * Kronecker(12, y). - Michael Somos, Feb 04 2015
G.f.: Product_{k>0} (1 - x^(3*k))^4 / ((1 - x^k) * (1 - x^(9*k))).
a(49*n + 4) = a(121*n + 10) = a(n).
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EXAMPLE
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G.f. = 1 + x + 2*x^2 - x^3 + x^4 - x^5 + x^6 - 3*x^7 - 2*x^8 + x^9 + ...
G.f. = q + q^13 + 2*q^25 - q^37 + q^49 - q^61 + q^73 - 3*q^85 - 2*q^97 + q^109 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ QPochhammer[ x^3]^4 / (QPochhammer[ x] QPochhammer[ x^9]), {x, 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^3 + A)^4 / (eta(x + A) * eta(x^9 + A)), n))};
(PARI) {a(n) = my(A, p, e, ap, w, k12, x, y, xy); if( n<0, 0, n = 12*n + 1; w = quadgen(12); (k12 = (u, v) -> kronecker( 12, u)*kronecker( 12, v)); (xy = (m) -> if(1, my(x); for(i=1, sqrtint( m\9), if( issquare( m - 9*i^2, &x), return([x, i]))))); A = factor(n); prod(k=1, matsize(A)[1], p = A[k, 1]; e = A[k, 2]; if( p<5, 0, p%12==7 || p%12==11, !(e%2), ap = if( p%12==5, [x, y] = xy(2*p); if(y%3==0, 0, w*k12(x, y)), [x, y] = xy(p); k12(9*y + x, y-x) + k12(9*y - x, y+x)); polchebyshev(e, 2, ap/2))))}; /* Michael Somos, Feb 04 2015 */
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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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