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A133563
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Expansion of chi(-q) / chi(-q^5) in powers of q where chi() is a Ramanujan theta function.
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0
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1, -1, 0, -1, 1, 0, 0, -1, 1, -1, 2, -2, 2, -2, 2, -1, 2, -3, 2, -3, 5, -5, 4, -5, 6, -4, 4, -7, 7, -7, 10, -11, 10, -12, 12, -10, 12, -15, 14, -16, 22, -22, 20, -24, 26, -22, 24, -30, 31, -33, 40, -43, 42, -46, 48, -45, 50, -58, 58, -63, 77, -79, 76, -86, 92, -86, 92, -107, 110, -116, 134, -141, 142, -154, 160, -157
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,11
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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 q^(-1/6) * eta(q) * eta(q^10) / ( eta(q^2) * eta(q^5) ) in powers of q.
Euler transform of period 10 sequence [ -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, ...].
G.f. is a period 1 Fourier series of a level 360 modular function which satisfies f(-1 / (360 t)) = f(t) where q = exp(2 pi i t).
Given g.f. A(x) then B(x) = x * A(x^6) satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = v * (u^2 - v) + w^2 * (u^2 + v).
Given g.f. A(x) then B(x) = x * A(x^6) satisfies 0 = f(B(x), B(x^3), B(x^9)) where f(u, v, w) = (u^3 + w^3) * (v + v^3) + 2 * v^4 - v^2 + u^3 * w^3 * ( 2 - v^2 ).
Given g.f. A(x) then B(x) = x * A(x^6) satisfies 0 = f(B(x), B(x^2), B(x^5), B(x^10)) where f(u1, u2, u5, u10) = u1^2 * u5^2 + u1^2 * u10^4 + u1 * u2^2 * u5 * u10^2 + u2 * u5^2 * u10^3 + u2^3 * u10^3 - u2^2 * u10^2 - u1^3 * u5^3 - u1^4 * u10^2 - u1^3 * u2^2 * u5 - u1^2 * u2 * u5^2 * u10.
G.f. is product k>0 P10(x^k) where P10 is 10th cyclotomic polynomial.
G.f.: Product_{k>0} (1 + x^(5*k)) / (1 + x^k).
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EXAMPLE
| q - q^7 - q^19 + q^25 - q^43 + q^49 - q^55 + 2*q^61 - 2*q^67 + 2*q^73 - ...
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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^10 + A) / eta(x^2 + A) / eta(x^5 + A), n))}
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CROSSREFS
| Sequence in context: A202146 A087010 A098220 * A104518 A114295 A004216
Adjacent sequences: A133560 A133561 A133562 * A133564 A133565 A133566
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Sep 16 2007
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