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A229802
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Expansion of q * f(-q, -q^4)^5 / f(-q)^3 in powers of q where f() is a Ramanujan theta function.
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4
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1, -2, 4, -3, 1, 2, -2, 0, 3, -2, 2, -2, 4, -6, 4, 1, -2, 4, 0, -3, 2, -4, 4, 0, 1, 2, 0, -4, 0, 2, 2, -2, 8, -6, -2, 1, -2, 0, 6, 0, 2, -4, 4, -6, 3, 2, -2, 4, -3, -2, 2, -2, 4, 0, 2, 0, 0, 0, 0, -2, 2, -4, 4, -3, 4, 4, -2, -4, 6, -6, 2, 0, 4, -6, 4, 0, -4
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OFFSET
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1,2
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COMMENTS
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LINKS
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FORMULA
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Expansion of q * f(-q)^2 * (f(-q^5) / f(-q^2, -q^3))^5 = q * f(-q, -q^4)^2 * (f(-q^5) / f(-q^2, -q^3))^3 in powers of q where f() is a Ramanujan theta function. - Michael Somos, Jun 10 2014
Euler transform of period 5 sequence [ -2, 3, 3, -2, -2, ...].
Moebius transform is period 5 sequence [ 1, -3, 3, -1, 0, ...]. - Michael Somos, Jun 10 2014
G.f.: x * (Product_{k>0} (1 - x^k)^2) / (Product_{k>0} (1 - x^(5*k - 2)) * (1 - x^(5*k - 3)))^5.
a(5*n) = a(n).
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EXAMPLE
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G.f. = q - 2*q^2 + 4*q^3 - 3*q^4 + q^5 + 2*q^6 - 2*q^7 + 3*q^9 - 2*q^10 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ q QPochhammer[ q]^2 / (QPochhammer[ q^2, q^5] QPochhammer[ q^3, q^5])^5, {q, 0, n}]; (* Michael Somos, Jun 10 2014 *)
a[ n_] := If[ n < 1, 0, Sum[ Im[(I - 3) {1, I, -I, -1, 0}[[ Mod[ d, 5, 1] ]] ], {d, Divisors @ n}]]; (* Michael Somos, Jun 10 2014 *)
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PROG
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(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( prod(k=1, n, (1 - x^k + A)^[ 2, 2, -3, -3, 2][k%5 + 1]), n))};
(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, imag( (I - 3) * [ 0, 1, I, -I, -1][ d%5 + 1])))}; /* Michael Somos, Jun 10 2014 */
(Sage) ModularForms( Gamma1(5), 1, prec=70).1;
(Magma) Basis( ModularForms( Gamma1(5), 1), 70) [2] ; /* Michael Somos, Jun 10 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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