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Search: id:A106459
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| A106459 |
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Expansion of q^(-1)eta(q^8)eta(q^32)/eta(q^16) in powers of q^8. |
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2
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| 1, -1, 0, -1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
(list; graph; listen)
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OFFSET
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0,1
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REFERENCES
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D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; p. 53, Exer. 2.2.10
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FORMULA
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Euler transform of period 4 sequence [ -1, 0, -1, -1, ...].
Given g.f. A(x), then B(x)=x*A(x^8) satisfies 0=f(B(x), B(x^2), B(x^3), B(x^6)) where f(u1, u2, u3, u6)=u1^4*u6^4 +u1^3*u2*u3^3*u6 +2*u1*u2^3*u3*u6^3 -u2^4*u3^4.
a(n)=b(8n+1) where b(n) is multiplicative and b(p^e) = kronecker(8, p)^(e/2) if e even, b(p^e) = 0 if e odd.
G.f.: 1/B(x) where B(x)= g.f. A006950.
Expansion of psi(-q)=f(-q,-q^3) in powers of q where f(a,b)=Sum_{k} a^((k^2+k)/2)*b^((k^2-k)/2) is Ramanujan's two-variable theta function.
G.f.: Product_{k>0} (1-x^k)*(1+x^(2*k)) = Product_{k>0} (1-x^k)*(1-x^(4*k-2)).
G.f.: Product_{k>0} (1-x^(2*k))/(1+x^(2*k-1)) = Product_{k>0} (1-x^(4*k))*(1-x^(2*k-1)).
Sum_{k>=0} a(k)x^(8k+1) = Sum_{k} (-1)^k x^((4k+1)^2).
G.f.: Sum_{k>=0} (-x)^(k*(k+1)/2) = Sum_{k} x^(8k^2+2k) -x^(8k^2+6k+1)
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EXAMPLE
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q - q^9 - q^25 + q^49 + q^81 - q^121 - q^169 + q^225 + q^289 -...
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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+A)*eta(x^4+A)/eta(x^2+A), n))}
(PARI) {a(n)=local(x); if(issquare(8*n+1, &x), kronecker(8, x))}
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CROSSREFS
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Sequence in context: A113430 A113681 A010054 this_sequence A033806 A033802 A033800
Adjacent sequences: A106456 A106457 A106458 this_sequence A106460 A106461 A106462
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KEYWORD
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sign
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AUTHOR
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Michael Somos, May 02 2005
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