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Search: id:A139093
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| A139093 |
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Expansion of eta(q^2)^7 / (eta(q)^2 * eta(q^4)^3) in powers of q. |
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+0
1
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| 1, 2, -2, -4, 2, 0, -4, 0, 2, 6, 0, -4, 4, 0, 0, 0, 2, 4, -6, -4, 0, 0, -4, 0, 4, 2, 0, -8, 0, 0, 0, 0, 2, 8, -4, 0, 6, 0, -4, 0, 0, 4, 0, -4, 4, 0, 0, 0, 4, 2, -2, -8, 0, 0, -8, 0, 0, 8, 0, -4, 0, 0, 0, 0, 2, 0, -8, -4, 4, 0, 0, 0, 6, 4, 0, -4, 4, 0, 0, 0, 0, 10, -4, -4, 0, 0, -4, 0, 4, 4, 0, 0, 0, 0, 0, 0, 4, 4, -2, -12, 2, 0, -8, 0
(list; graph; listen)
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OFFSET
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0,2
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FORMULA
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Expansion of phi(q) * phi(-q^2) in powers of q where phi() is a Ramanujan theta function.
Euler transform of period 4 sequence [ 2, -5, 2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 8^(1/2) (t/i) g(t) where q = exp(2 pi i t) and g() is g.f. for A112603.
a(8*n + 5) = a(8*n + 7) = 0.
G.f.: Product_{k>0} (1 - x^(2*k))^2 * (1 + x^(2*k-1))^2 / (1 + x^(2*k)).
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EXAMPLE
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1 + 2*q - 2*q^2 - 4*q^3 + 2*q^4 - 4*q^6 + 2*q^8 + 6*q^9 - 4*q^11 + ...
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PROGRAM
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(PARI) {a(n) = if( n<1, n==0, 2 * (-1)^(n\2) * sumdiv(n, d, kronecker(-8, d)))}
(PARI) {a(n) = local(A); if ( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^7 / eta(x + A)^2 / eta(x^4 + A)^3, n))}
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CROSSREFS
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Cf. A082564(n) = (-1)^n * a(n). A133692(n) = a(2*n). 2 * A125095(n) = a(2*n + 1). A033715(n) = a(4*n). 2 * A112603(n) = a(8*n). -4 * A033761(n) = a(8*n+3).
Sequence in context: A033715 A082564 A133692 this_sequence A080918 A033758 A033750
Adjacent sequences: A139090 A139091 A139092 this_sequence A139094 A139095 A139096
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KEYWORD
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sign
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AUTHOR
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Michael Somos, Apr 08 2008
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