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A275578
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Expansion of (F(x) + F(-x)) / 2 in powers of x^2 where F(x) = (f(-x) * f(-x^11))^2 and f() is a Ramanujan theta function.
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1
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1, -1, 1, -2, -2, 1, 4, -1, -2, 0, 2, -1, -4, 5, 0, 7, -1, -2, 3, -4, -8, -6, -2, 8, -3, 2, -6, 1, 0, 5, 12, 4, 4, -7, 1, -3, 4, 4, -2, -10, 1, -6, -2, 0, 15, -8, -7, 0, -7, -2, 2, -16, 2, 18, 10, -3, 9, -1, -8, 4, 1, 8, -9, 8, 6, -18, 0, 5, -7, 10, -8, 4, 0
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OFFSET
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0,4
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COMMENTS
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Fourier expansion of a multiplicative cusp form on Gamma_0(44).
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LINKS
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FORMULA
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Expansion of F(x) + 2*x*F(x^2) + 2*x^3*F(x^4) in powers of x^2 where F(x) = (f(-x) * f(-x^11))^2 and f() is a Ramanujan theta function.
G.f. is a period 1 Fourier series which satisfies f(-1 / (44 t)) = 22 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A279371.
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EXAMPLE
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G.f. = 1 - x + x^2 - 2*x^3 - 2*x^4 + x^5 + 4*x^6 - x^7 - 2*x^8 + 2*x^10 + ...
G.f. = q - q^3 + q^5 - 2*q^7 - 2*q^9 + q^11 + 4*q^13 - q^15 - 2*q^17 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^11])^2, {x, 0, 2 n}];
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PROG
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(PARI) {a(n) = my(A); if( n<0, 0, n = 2*n; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^11 + A))^2, n))};
(Magma) A := Basis( CuspForms( Gamma0(44), 2), 146); A[1] - A[3];
(Sage) A = CuspForms( Gamma0(44), 2, prec=146) . basis(); A[0] - A[2];
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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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