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Search: id:A138532
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| A138532 |
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Expansion of psi(q) / psi(q^5) in powers of q where psi() is a Ramanujan theta function. |
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3
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| 1, 1, 0, 1, 0, -1, 0, 0, -1, 0, 2, 0, 0, 1, 0, -2, -1, 0, -2, 0, 3, 2, 0, 3, 0, -5, -2, 0, -3, 0, 6, 2, 0, 4, 0, -8, -3, 0, -6, 0, 11, 5, 0, 8, 0, -14, -6, 0, -10, 0, 18, 6, 0, 12, 0, -22, -9, 0, -16, 0, 28, 13, 0, 21, 0, -36, -14, 0, -25, 0, 44, 16, 0, 30, 0, -54, -22, 0, -38, 0, 67, 28, 0, 48, 0, -83, -32, 0, -57, 0, 100
(list; graph; listen)
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OFFSET
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0,11
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FORMULA
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Expansion of q^(1/2) * (eta(q^2) / eta(q^10))^2 * eta(q^5) / eta(q) in powers of q.
Euler transform of period 10 sequence [ 1, -1, 1, -1, 0, -1, 1, -1, 1, 0, ...].
Given g.f. A(x), then B(x) = A(x^2) / x satisfies 0 = f(B(x), B(x^2)) where f(u, v) = (v^2 - u^2)^2 - (u^2 - 1) * (u^2 - 5) * v^2.
Given g.f. A(x), then B(x) = A(x^2) / x satisfies 0 = f(B(x), B(x^3)) where f(u, v) = (v^2 - u^2) * (u + v)^2 - u * v * (u^2 - 1) * (v^2 - 5).
Given g.f. A(x), then B(x) = A(x^2) / x satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = u^2 * w * (v^2 - 1) - v * (v + w)^2.
Given g.f. A(x), then B(x) = A(x^2) / x satisfies 0 = f(B(x), B(x^2), B(x^3), B(x^6)) where f(u1, u2, u3, u6) = (u1 * u6 - u2 * u3)^2 - u2 * u6 * (u3^2 - u1^2).
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 A138526.
a(5*n + 2) = a(5*n + 4) = 0.
G.f.: (Product_{k>0} P(5,x^k) * P(10,x^k)^2)^(-1) where P(n,x) is the nth cyclotomic polynomial
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EXAMPLE
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1/q + q + q^5 - q^9 - q^15 + 2*q^19 + q^25 - 2*q^29 - q^31 - 2*q^35 + ...
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PROGRAM
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(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^5 + A) / eta(x + A) * (eta(x^2 + A) / eta(x^10 + A))^2, n))}
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CROSSREFS
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Convolution square is A138516. Convolution inverse is A116494.
Sequence in context: A076544 A084143 A025888 this_sequence A065293 A054876 A109502
Adjacent sequences: A138529 A138530 A138531 this_sequence A138533 A138534 A138535
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KEYWORD
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sign
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AUTHOR
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Michael Somos, Mar 23 2008
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