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A145977
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Expansion of c(q^3) / (c(q^3) + c(q^6)) where c() is a cubic AGM function.
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0
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1, -1, 1, -1, 2, -3, 4, -5, 7, -10, 12, -15, 20, -26, 32, -39, 50, -63, 76, -92, 114, -140, 168, -201, 244, -295, 350, -415, 496, -591, 696, -818, 967, -1140, 1332, -1554, 1820, -2126, 2468, -2861, 3324, -3855, 4448, -5126, 5916, -6816, 7824, -8970, 10292, -11793, 13471, -15372, 17548, -20007
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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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 1 - q * psi(q^9) / psi(q) = phi(-q^9) / (psi(q) * chi(-q^3)) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of eta(q) * eta(q^6) * eta(q^9)^2 / (eta(q^2)^2 * eta(q^3) * eta(q^18)), in powers of q.
Euler transform of period 18 sequence [ -1, 1, 0, 1, -1, 1, -1, 1, -2, 1, -1, 1, -1, 1, 0, 1, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1/(18 t)) = (2/3) / f(t) where q = exp(2 pi i t).
G.f.: Product_{k>0} (P(3, x^k) * P(9, x^k)) / (P(4, x^k)^2 * P(18, x^k)) where P(n, x) is n-th cyclotomic polynomial.
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EXAMPLE
| 1 - q + q^2 - q^3 + 2*q^4 - 3*q^5 + 4*q^6 - 5*q^7 + 7*q^8 - 10*q^9 + ...
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PROG
| (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A) * eta(x^9 + A)^2 / (eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^18 + A)), n))}
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CROSSREFS
| Convolution inverse of A139032. - A124243(n) = a(n) unless n=0. A128129(n) = a(2*n) unless n=0. - A132302(n) = a(2*n + 1). A128641(n) = a(3*n).
Sequence in context: A036033 A124243 A132975 * A050729 A117536 A104665
Adjacent sequences: A145974 A145975 A145976 * A145978 A145979 A145980
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Oct 26 2008
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