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A139380
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Expansion of phi(q) / phi(q^9) in powers of q where phi() is a Ramanujan theta function.
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1
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1, 2, 0, 0, 2, 0, 0, 0, 0, 0, -4, 0, 0, -4, 0, 0, 2, 0, 0, 8, 0, 0, 8, 0, 0, -2, 0, 0, -16, 0, 0, -16, 0, 0, 4, 0, 0, 28, 0, 0, 28, 0, 0, -8, 0, 0, -48, 0, 0, -46, 0, 0, 12, 0, 0, 80, 0, 0, 76, 0, 0, -20, 0, 0, -126, 0, 0, -120, 0, 0, 32, 0, 0, 196, 0, 0, 184, 0, 0, -48, 0, 0, -300, 0, 0, -280, 0, 0, 72, 0, 0
(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 1 + 2 * q * chi(q^3) / chi(q^9)^3 in powers of q where chi() is a Ramanujan theta function.
Expansion of 1 - 2 * c(q^6) / c(-q^3) in powers of q where c() is a cubic AGM function.
Expansion of eta(q^2)^5 * eta(q^9)^2 * eta(q^36)^2 / (eta(q)^2 * eta(q^4)^2 * eta(q^18)^5) in powers of q.
Euler transform of period 36 sequence [ 2, -3, 2, -1, 2, -3, 2, -1, 0, -3, 2, -1, 2, -3, 2, -1, 2, 0, 2, -1, 2, -3, 2, -1, 2, -3, 0, -1, 2, -3, 2, -1, 2, -3, 2, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u - v)^3 - u * (3 - u) * (v - 1) * (3 - 2*u + u*v).
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 3 / f(t) where q = exp(2 pi i t).
a(3*n) = 0 unless n=0. a(3*n + 2) = 0.
G.f.: (1 + 2 * Sum_{k>0} x^k^2) / (1 + 2 * Sum_{k>0} x^(9*k^2)).
G.f.: Product_{k>0} (1 - x^(2*k)) * (1 + x^(2*k-1))^2 / ((1 - x^(18*k)) * (1 + x^(18*k-9))^2).
A128771(n) = (-1)^n * a(n). 2 * A128111(n) = a(3*n + 1).
Empirical : sum(exp(-Pi/3)^(n-1)*a(n),n=1..infinity) = sqrt(3). Simon Plouffe, Feb. 20, 2011.
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EXAMPLE
| 1 + 2*q + 2*q^4 - 4*q^10 - 4*q^13 + 2*q^16 + 8*q^19 + 8*q^22 - 2*q^25 + ...
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PROG
| (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^9 + A)^2 * eta(x^36 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^18 + A)^5), n))}
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CROSSREFS
| Sequence in context: A193531 A093492 A128771 * A000122 A002448 A033759
Adjacent sequences: A139377 A139378 A139379 * A139381 A139382 A139383
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Apr 15 2008
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