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A133573
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Expansion of ( 5 * phi(-q^5)^2 - phi(-q)^2 ) / 4 in powers of q where phi() is a Ramanujan theta function.
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2
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1, 1, -1, 0, -1, -3, 0, 0, -1, 1, 3, 0, 0, 2, 0, 0, -1, 2, -1, 0, 3, 0, 0, 0, 0, -7, -2, 0, 0, 2, 0, 0, -1, 0, -2, 0, -1, 2, 0, 0, 3, 2, 0, 0, 0, -3, 0, 0, 0, 1, 7, 0, -2, 2, 0, 0, 0, 0, -2, 0, 0, 2, 0, 0, -1, -6, 0, 0, -2, 0, 0, 0, -1, 2, -2, 0, 0, 0, 0, 0, 3, 1, -2, 0, 0, -6, 0, 0, 0, 2, 3, 0, 0, 0, 0, 0, 0, 2, -1, 0, 7, 2, 0, 0, -2
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,6
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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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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| Expansion of eta(q^2)^3 * eta(q^5) / ( eta(q) * eta(q^10) ) in powers of q.
Euler transform of period 10 sequence [ 1, -2, 1, -2, 0, -2, 1, -2, 1, -2, ...].
Moebius transform is period 40 sequence [ 1, -2, -1, 0, -4, 2, -1, 0, 1, 8, -1, 0, 1, 2, 4, 0, 1, -2, -1, 0, 1, 2, -1, 0, -4, -2, -1, 0, 1, -8, -1, 0, 1, -2, 4, 0, 1, 2, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (40 t)) = 20 (t/i) g(t) where q = exp(2 pi i t) and g() is g.f. for A122190.
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EXAMPLE
| 1 + q - q^2 - q^4 - 3*q^5 - q^8 + q^9 + 3*q^10 + 2*q^13 - q^16 + ...
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PROG
| (PARI) {a(n) = if( n<1, n==0, (-1)^n * sumdiv(n, d, if( d%5==0, kronecker(-4, d/5) * 5) - kronecker(-4, d)))}
(PARI) {a(n) = local(A); if( n<0, 0, A = x*O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^5+A) / eta(x + A) / eta(x^10 + A), n))}
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CROSSREFS
| A133574(n) = (-1)^n * a(n).
Sequence in context: A100655 A079275 * A133574 A151859 A163541 A165974
Adjacent sequences: A133570 A133571 A133572 * A133574 A133575 A133576
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Sep 17 2007
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