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Search: id:A115978
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| A115978 |
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Expansion of theta_4(q)theta_4(q^3) in powers of q. |
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+0
3
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| 1, -2, 0, -2, 6, 0, 0, -4, 0, -2, 0, 0, 6, -4, 0, 0, 6, 0, 0, -4, 0, -4, 0, 0, 0, -2, 0, -2, 12, 0, 0, -4, 0, 0, 0, 0, 6, -4, 0, -4, 0, 0, 0, -4, 0, 0, 0, 0, 6, -6, 0, 0, 12, 0, 0, 0, 0, -4, 0, 0, 0, -4, 0, -4, 6, 0, 0, -4, 0, 0, 0, 0, 0, -4, 0, -2, 12, 0, 0, -4, 0, -2, 0, 0, 12, 0, 0, 0, 0, 0, 0, -8, 0, -4, 0, 0, 0, -4, 0, 0, 6, 0
(list; graph; listen)
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OFFSET
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0,2
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FORMULA
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Euler transform of period 6 sequence [ -2,-1,-4,-1,-2,-2,...].
Expansion of (eta(q)*eta(q^3))^2/(eta(q^2)*eta(q^6)) in powers of q.
a(n)=-2*b(n) where b(n) is multiplicative and b(2^e) = -3(1+(-1)^e)/2 if e>0, b(3^e)=1, b(p^e) = 1+e if p == 1 (mod 6), b(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6).
Given g.f. A(x), then B(x)=A(x)^2 satisfies 0=f(B(x),B(x^2),B(x^4)) where f(u,v,w)=v*(u+v)^2-4*u*(w^2-v*w+v^2).
G.f.: 1 -2(Sum_{k>0} x^(k)/(1+x^k+x^(2k)) -4x^(4k)/(1+x^(4k)+x^(8k))).
G.f.: theta_4(q)theta_4(q^3) = (Sum_{k} (-x)^(k^2))(Sum_{k} (-x)^(3k^2))
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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^3+A))^2/eta(x^2+A)/eta(x^6+A), n))}
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CROSSREFS
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a(n)=-2*A115979(n) if n>0. a(n)=(-1)^n*A033716(n).
Sequence in context: A059432 A113772 A033716 this_sequence A033751 A033745 A033721
Adjacent sequences: A115975 A115976 A115977 this_sequence A115979 A115980 A115981
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KEYWORD
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sign
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AUTHOR
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Michael Somos, Feb 09 2006
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