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A133690
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Expansion of ( phi(-q) * phi(q^2) )^2 in powers of q where phi() is a Ramanujan theta function.
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3
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1, -4, 8, -16, 24, -24, 32, -32, 24, -52, 48, -48, 96, -56, 64, -96, 24, -72, 104, -80, 144, -128, 96, -96, 96, -124, 112, -160, 192, -120, 192, -128, 24, -192, 144, -192, 312, -152, 160, -224, 144, -168, 256, -176, 288, -312, 192, -192, 96, -228, 248, -288, 336, -216, 320, -288, 192, -320, 240
(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 * eta(q^4)^5 / ( eta(q^2)^3 * eta(q^8)^2 ) )^2 in powers of q.
Euler transform of period 8 sequence [ -4, 2, -4, -8, -4, 2, -4, -4, ...].
a(n) = -4 * b(n) where b(n) is multiplicative with b(2) = -2, b(2^e) = -6 if e>1, b(p^e) = (p^(e+1) - 1) / (p - 1) if p>2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 32 (t/i)^2 g(t) where q = exp(2 pi i t) and g() is g.f. for A133657.
G.f.: ( Product_{k>0} (1 - x^k)^2 * (1 + x^(2*k))^3 / (1 + x^(4*k))^2 )^2.
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EXAMPLE
| 1 - 4*q + 8*q^2 - 16*q^3 + 24*q^4 - 24*q^5 + 32*q^6 - 32*q^7 + 24*q^8 - ...
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PROG
| (PARI) {a(n) = if( n<1, n==0, -4 * if( n%2, sigma(n), if( n%4, -2 * sigma(n/2), -6 * sumdiv( n/4, d, (d%2)*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 )^2, n))}
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CROSSREFS
| Convolution square of A133692. (-1)^n * A097057(n) = a(n). 8 * A046897(n) = a(2*n) unless n=0. -4 * A008438(n) = a(2*n+1). A004011(n) = a(4*n). -4 * A112610(n) = a(4*n+1). -16 * A097723(n) = a(4*n+3).
Sequence in context: A053688 A036302 A032377 * A097057 A160746 A160740
Adjacent sequences: A133687 A133688 A133689 * A133691 A133692 A133693
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Sep 20 2007
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