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A132973
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Expansion of psi(-q)^3 / psi(-q^3) in powers of q where psi() is a Ramanujan theta function.
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4
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1, -3, 3, -3, 3, 0, 3, -6, 3, -3, 0, 0, 3, -6, 6, 0, 3, 0, 3, -6, 0, -6, 0, 0, 3, -3, 6, -3, 6, 0, 0, -6, 3, 0, 0, 0, 3, -6, 6, -6, 0, 0, 6, -6, 0, 0, 0, 0, 3, -9, 3, 0, 6, 0, 3, 0, 6, -6, 0, 0, 0, -6, 6, -6, 3, 0, 0, -6, 0, 0, 0, 0, 3, -6, 6, -3, 6, 0, 6, -6, 0, -3, 0, 0, 6, 0, 6, 0, 0, 0, 0, -12, 0, -6, 0, 0, 3, -6, 9, 0, 3, 0, 0, -6
(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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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| Expansion of b(q^2)^2 / b(-q) in powers of q where b() is a cubic AGM function.
Expansion of eta(q)^3 * eta(q^4)^3 * eta(q^6) / ( eta(q^2)^3 * eta(q^3) * eta(q^12) ) in powers of q.
Euler transform of period 12 sequence [ -3, 0, -2, -3, -3, 0, -3, -3, -2, 0, -3, -2, ...].
Moebius transform is period 12 sequence [ -3, 6, 0, 0, 3, 0, -3, 0, 0, -6, 3, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 108^(1/2) (t/i) g(t) where q = exp(2 pi i t) and g(t) is g.f. for A113447.
a(6*n+5) = 0.
G.f.: Product_{k>0} (1 - x^k)^3 * (1 + x^(2*k))^3 / ((1 - x^(3*k)) * (1 + x^(6*k))).
G.f.: 1 + 3 * Sum_{k>0} (-1)^k * (x^k + x^(3*k)) / (1 + x^k + x^(2*k)).
G.f.: 1 + 3 * ( Sum_{k>0} x^(6*k-5) / ( 1 + x^(6*k-5) ) - x^(6*k-1) / ( 1 + x^(6*k-1) )).
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EXAMPLE
| 1 - 3*q + 3*q^2 - 3*q^3 + 3*q^4 + 3*q^6 - 6*q^7 + 3*q^8 - 3*q^9 + 3*q^12 + ...
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PROG
| (PARI) {a(n) = if( n<1, n==0, 3 * (-1)^n * sumdiv(n, d, kronecker(-12, d)))}
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^6 + A) / ( eta(x^2 + A)^3 * eta(x^3 + A) * eta(x^12 + A ) ), n))}
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CROSSREFS
| (-1)^n * A107760(n) = a(n). Convolution inverse of A132974.
Sequence in context: A200777 A033700 A122916 * A107760 A180560 A172368
Adjacent sequences: A132970 A132971 A132972 * A132974 A132975 A132976
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Sep 07 2007
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