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Search: id:A133988
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| A133988 |
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Expansion of phi(q) / chi(q^3) in powers of q where phi(), chi() are Ramanujan theta functions. |
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+0
2
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| 1, 2, 0, -1, 0, 0, 1, 0, 0, 0, -2, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
(list; graph; listen)
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OFFSET
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0,2
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FORMULA
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Expansion of psi(-q) + 3 * q * psi(-q^9) in powers of q where psi() is a Ramanujan theta function.
Expansion of q^(-1/8) eta(q^2)^5 * eta(q^3) * eta(q^12) / ( eta(q) * eta(q^4) * eta(q^6) )^2 in powers of q.
Euler transform of period 12 sequence [ 2, -3, 1, -1, 2, -2, 2, -1, 1, -3, 2, -1, ...].
a(n) = b(8*n+1) where b(n) is multiplicative and b(3^(2e)) = -2 * (-1)^e, b(p^(2e)) = (-1)^e if p == 3, 5 (mod 8), b(p^(2e)) = +1 if p == 1, 7 (mod 8) and b(p^(2e-1)) = b(2^e) = 0 if e>0.
G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = 12 (t/i)^(1/2) g(t) where q = exp(2 pi i t) and g(t) is g.f. for A133985.
G.f.: Sum_{k>0} (-1)^[k/2] * (x^((k^2-k)/2) + 3 * x^(9*(k^2-k)/2+1) ).
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EXAMPLE
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q + 2*q^9 - q^25 + q^49 - 2*q^81 - q^121 - q^169 - 2*q^225 + q^289 - ...
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PROGRAM
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(PARI) {a(n) = (-1)^(n\3) * ((n+1)%3) * issquare( 8*n+1)}
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A) * eta(x^12 + A) / ( eta(x + A) * eta(x^4 + A) * eta(x^6 + A) )^2, n))}
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CROSSREFS
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A089812(n) = (-1)^n * a(n).
Sequence in context: A124304 A165408 A089812 this_sequence A123858 A035145 A107064
Adjacent sequences: A133985 A133986 A133987 this_sequence A133989 A133990 A133991
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KEYWORD
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sign
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AUTHOR
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Michael Somos, Oct 01 2007
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