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A164272
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Expansion of phi(q) * phi(-q^3) in powers of q where phi() is a Ramanujan theta function.
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2
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1, 2, 0, -2, -2, 0, 0, -4, 0, 2, 0, 0, -2, 4, 0, 0, 6, 0, 0, -4, 0, 4, 0, 0, 0, 2, 0, -2, -4, 0, 0, -4, 0, 0, 0, 0, -2, 4, 0, -4, 0, 0, 0, -4, 0, 0, 0, 0, 6, 6, 0, 0, -4, 0, 0, 0, 0, 4, 0, 0, 0, 4, 0, -4, 6, 0, 0, -4, 0, 0, 0, 0, 0, 4, 0, -2, -4, 0, 0, -4, 0, 2, 0, 0, -4, 0, 0, 0, 0, 0, 0, -8, 0, 4, 0, 0, 0, 4, 0, 0, -2, 0, 0, -4, 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 eta(q^2)^5 * eta(q^3)^2 / (eta(q)^2 * eta(q^4)^2 * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [ 2, -3, 0, -1, 2, -4, 2, -1, 0, -3, 2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 768^(1/2) (t/i) g(t) where q = exp(2 pi i t) and g() is g.f. for A112604.
a(4*n + 2) = 0.
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EXAMPLE
| 1 + 2*q - 2*q^3 - 2*q^4 - 4*q^7 + 2*q^9 - 2*q^12 + 4*q^13 + 6*q^16 + ...
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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^3 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^6 + A)), n))}
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CROSSREFS
| Cf. (-1)^n * A164273(n) = a(n). 2 * A129449(n) = a(2*n + 1). A115978(n) = a(4*n). 2 * A112604(n) = a(4*n + 1). -2 * A112605(n) = a(4*n + 3).
Sequence in context: A091379 A151758 * A164273 A106277 A088627 A024713
Adjacent sequences: A164269 A164270 A164271 * A164273 A164274 A164275
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Aug 11 2009
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