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A133079
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Expansion of f(x)^3 - 3 * x * f(x^9)^3 in powers of x^3 where f() is a Ramanujan theta function.
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5
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1, -5, -7, 0, 0, 11, 0, -13, 0, 0, 0, 0, 17, 0, 0, 19, 0, 0, 0, 0, 0, 0, -23, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0, -29, 0, 0, 0, 0, -31, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 35, 0, 0, 0, 0, 0, -37, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 41, 0, 0, 0, 0, 0, 0, 43, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -47, 0, 0, 0
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OFFSET
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0,2
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COMMENTS
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There is a plus sign on the left side and the first and third plus signs on the right side which should be minuses in Ramanujan's equation.
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REFERENCES
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B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 357, Entry 5, Eq. (5.1)
S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 1, see page 266.
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LINKS
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FORMULA
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Expansion of f(x) * a(-x) in powers of x where f() is a Ramanujan theta function and a() is a cubic AGM theta function.
G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = -192 (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A204850.
a(n) = b(24*n + 1) where b(n) is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * p^(e/2) if p == 1, 3 (mod 8), b(p^e) = (1 + (-1)^e)/2 * (-p)^(e/2) if p == 5, 7 (mod 8).
G.f.: Sum_{k in Z} Kronecker( 2, 2*k + 1) * (6*k + 1) * x^(k * (3*k + 1)/2).
a(5*n + 3) = a(5*n + 4) = 0. a(25*n + 1) = -5 * a(n). a(n) = (-1)^n * A116916(n).
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EXAMPLE
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G.f. = 1 - 5*x - 7*x^2 + 11*x^5 - 13*x^7 + 17*x^12 + 19*x^15 - 23*x^22 + ...
G.f. = q - 5*q^25 - 7*q^49 + 11*q^121 - 13*q^169 + 17*q^289 + 19*q^361 - ...
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MATHEMATICA
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a[ n_] := With[ {m = Sqrt[24 n + 1]}, If[ IntegerQ@m, m (-1)^Boole[Mod[m, 8] > 4], 0]]; (* Michael Somos, Jun 19 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x]^3 - 3 x QPochhammer[ -x^9]^3, {x, 0, 3 n}]; (* Michael Somos, Jun 19 2015 *)
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PROG
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(PARI) {a(n) = if( issquare( 24*n + 1, &n), n * (-1) ^ (n%8 > 4), 0)};
(PARI) {a(n) = my(A, p, e); if( n<0, 0, n = 24*n + 1; A = factor(n); prod(k = 1, matsize(A) [1], [p, e] = A[k, ]; if( p < 5, 0, p *= kronecker( -2, p); if( e%2, 0, p^(e/2) ))))};
(PARI) {a(n) = my(A); if( n<0, 0, n *= 3; A = x * O(x^n); polcoeff( eta(-x + A)^3 - 3 * x * eta(-x^9 + A)^3, 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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