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A138527
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Expansion of phi(-q) / phi(-q^5) in powers of q where phi() is a Ramanujan theta function.
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4
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1, -2, 0, 0, 2, 2, -4, 0, 0, 2, 4, -8, 0, 0, 4, 8, -14, 0, 0, 8, 14, -24, 0, 0, 12, 22, -40, 0, 0, 20, 36, -64, 0, 0, 32, 56, -98, 0, 0, 48, 84, -148, 0, 0, 72, 126, -220, 0, 0, 106, 184, -320, 0, 0, 152, 264, -460, 0, 0, 216, 376, -652, 0, 0, 306, 528, -912, 0, 0, 424, 732, -1264, 0, 0, 584, 1008
(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) / eta(q^5))^2 * eta(q^10) / eta(q^2) in powers of q.
Euler transform of period 10 sequence [ -2, -1, -2, -1, 0, -1, -2, -1, -2, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (v^2 - u^2)^2 - u^2 * (1 - v^2) * (5 - v^2).
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (v^2 - u^2) * (u + v)^2 - u * v * (1 - u^2) * (5 - v^2).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u + v)^2 * w^2 - u * v * (5 - v^2).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = (u2 * u3 - u1 * u6)^2 - u1 * u3 * (u6^2 - u2^2).
G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = 5^(1/2) g(t) where q = exp(2 pi i t) and g() is g.f. for A116494.
G.f.: Product_{k>0} P(10, x^k) / P(5, x^k) where P(n, x) is the n-th cyclotomic polynomial.
a(5*n + 2) = a(5*n + 3) = 0.
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EXAMPLE
| 1 - 2*q + 2*q^4 + 2*q^5 - 4*q^6 + 2*q^9 + 4*q^10 - 8*q^11 + 4*q^14 + ...
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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^5 + A))^2 * eta(x^10 + A) / eta(x^2 + A), n))}
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CROSSREFS
| Convolution inverse is A138526. Convolution square is A138518.
Sequence in context: A146165 A048866 * A144377 A033718 A033737 A033747
Adjacent sequences: A138524 A138525 A138526 * A138528 A138529 A138530
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Mar 23 2008
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