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A134343
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Expansion of psi(-q)^2 in powers of q where psi() is a Ramanujan theta function.
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1
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1, -2, 1, -2, 2, 0, 3, -2, 0, -2, 2, -2, 1, -2, 0, -2, 4, 0, 2, 0, 1, -4, 2, 0, 2, -2, 0, -2, 2, -2, 1, -4, 0, 0, 2, 0, 4, -2, 2, -2, 0, 0, 3, -2, 0, -2, 4, 0, 2, -2, 0, -4, 0, 0, 0, -4, 3, -2, 2, 0, 2, -2, 0, 0, 2, -2, 4, -2, 0, -2, 2, 0, 3, -2, 0, 0, 4, 0, 2, -2, 0, -6, 0, -2, 2, 0, 0, -2, 2, 0, 1, -4, 2, -2, 4, 0, 0, -2, 0, -2, 2, -2, 2, 0, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
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REFERENCES
| Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| Expansion of q^(-1/4) * (eta(q) * eta(q^4) / eta(q^2))^2 in powers of q.
Euler transform of period 4 sequence [ -2, 0, -2, -2, ...].
G.f. is Fourier series which satisfies f(-1/ (64 t)) = 8 (t/i) f(t) where q = exp(2 pi i t).
a(n) = b(4*n+1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4), b(p^e) = e + 1 if p == 1 (mod 8), b(p^e) = (-1)^e * (e + 1) if p == 5 (mod 8).
a(9*n+5) = a(9*n+8) = 0.
G.f.: (Product_{k>0} (1 - x^k) * (1 + x^(2*k)))^2.
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EXAMPLE
| q - 2*q^5 + q^9 - 2*q^13 + 2*q^17 + 3*q^25 - 2*q^29 - 2*q^37 + 2*q^41 - ...
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PROG
| (PARI) {a(n) = if( n<0, 0, (-1)^n * sumdiv(4 * n + 1, d, (-1)^(d\2)))}
(PARI) {a(n) = local(A); if( n<0, 0, A = x*O(x^n); polcoeff( ( eta(x + A) * eta(x^4 + A) / eta(x^2 + A) )^2, n))}
(PARI)
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CROSSREFS
| (-1)^n * A008441(n) = a(n). A113407(n) = a(2*n). -2 * A053692(n) = a(2*n+1).
Sequence in context: A137579 A108805 A008441 * A108804 A200227 A127249
Adjacent sequences: A134340 A134341 A134342 * A134344 A134345 A134346
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Oct 21 2007
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