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A133692
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Expansion of phi(-q) * phi(q^2) in powers of q where phi() is a Ramanujan theta function.
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4
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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; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
a(n) is nonzero if and only if n is in A002479.
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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 * eta(q^4)^5 / ( eta(q^2)^3 * eta(q^8)^2 ) in powers of q.
Euler transform of period 8 sequence [ -2, 1, -2, -4, -2, 1, -2, -2, ...].
Moebius transform is period 16 sequence [ -2, 4, -2, 0, 2, 4, 2, 0, -2, -4, -2, 0, 2, -4, 2, 0, ...].
a(n) = -2 * b(n) where b(n) is multiplicative with b(2^e) = -1 if e>0, b(p^e) = (1 + (-1)^e) / 2 if p == 5, 7 (mod 8), b(p^e) = e + 1 if p == 1, 3 (mod 8).
G.f.: Product_{k>0} (1 - x^k)^2 * (1 + x^(2*k))^3 / (1 + x^(4*k))^2.
a(8*n + 5) = a(8*n + 7) = 0.
A133690 is convolution square. (-1)^n * A033715(n) = a(n). A033715(n) = a(2*n). -2 * A113411(n) = a(2*n+1).
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EXAMPLE
| 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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PROG
| (PARI) {a(n) = if( n<1, n==0, (-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 + A)^2 * eta(x^4 + A)^5 / eta(x^2 + A)^3 / eta(x^8 + A)^2, n))}
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CROSSREFS
| Cf. A002479, A033715, A113441, A133690.
Sequence in context: A033715 A082564 * A139093 A080918 A033758 A033750
Adjacent sequences: A133689 A133690 A133691 * A133693 A133694 A133695
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Sep 20 2007
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