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A137569
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Expansion of q^(1/12) eta(q) / eta(q^3) in powers of q.
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2
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1, -1, -1, 1, -1, 0, 2, -1, -1, 3, -2, -1, 4, -3, -2, 5, -4, -2, 8, -6, -4, 10, -7, -4, 14, -10, -6, 18, -13, -7, 24, -17, -10, 30, -21, -12, 40, -28, -17, 49, -35, -19, 63, -44, -26, 78, -55, -31, 98, -69, -40, 120, -84, -47, 150, -105, -61, 182, -127, -71, 224, -156, -90, 271, -189, -106, 330, -229, -131, 396, -275
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,7
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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 f(-q) / f(-q^3) in powers of q where f() is a Ramanujan theta function.
Euler transform of period 3 sequence [ -1, -1, 0, ...].
Given g.f. A(x) then B(x) = A(x^6)^2 / x satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = 4*v^2 + (u^2 - v) * (w^2 + v).
G.f. is a period 1 Fourier series which satisfies f(-1 / (432 t)) = 3^(1/2) / f(t) where q = exp(2 pi i t).`
G.f.: Product_{k>0} (1 - x^(3*k-1)) * (1 - x^(3*k-2)).
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EXAMPLE
| q^-1 - q^11 - q^23 + q^35 - q^47 + 2*q^71 - q^83 - q^95 + 3*q^107 + ...
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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^3 + A), n))}
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CROSSREFS
| A035943(n) = a(3*n). -A035941(n) = a(3*n+1). -A035940(n) = a(3*n+2). Convolution inverse of A000726.
Sequence in context: A093394 A094363 A124832 * A089177 A023996 A049998
Adjacent sequences: A137566 A137567 A137568 * A137570 A137571 A137572
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Jan 26 2008
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