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Search: id:A125095
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| A125095 |
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Expansion of phi(-q)psi(q^4) in powers of q where psi(), phi() are Ramanujan theta functions. |
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2
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| 1, -2, 0, 0, 3, -2, 0, 0, 2, -2, 0, 0, 1, -4, 0, 0, 4, 0, 0, 0, 2, -2, 0, 0, 1, -4, 0, 0, 4, -2, 0, 0, 0, -2, 0, 0, 2, -2, 0, 0, 5, -2, 0, 0, 2, 0, 0, 0, 2, -6, 0, 0, 0, -2, 0, 0, 2, 0, 0, 0, 3, -4, 0, 0, 4, -2, 0, 0, 2, -2, 0, 0, 0, -2, 0, 0, 6, 0, 0, 0, 0, -2, 0, 0, 1, -6, 0, 0, 4, -2, 0, 0, 0, -4, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 4
(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 q^(-1/2)(eta(q)^2*eta(q^8)^2)/(eta(q^2)*eta(q^4)) in powers of q.
Given g.f. A(x), then B(x)=x*A(x^2) satisfies 0=f(B(x), B(x^2), B(x^3), B(x^6)) where f(u1, u2, u3, u6)= +u1^2*u6*(u1+3*u3) +2*u2^2*u3*(u2+3*u6) -3*u3^2*u2*(u1+u3) -6*u6^2*u1*(u2+u6).
a(n) = b(2n+1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (e+1)(-1)^e if p == 1, 3 (mod 8), b(p^e) = (1+(-1)^e)/2 if p == 5, 7 (mod 8).
Euler transform of period 8 sequence [ -2, -1, -2, 0, -2, -1, -2, -2, ...].
a(4n+2)=a(4n+3)=0.
G.f.: (Sum_{k} (-1)^k*x^k^2)(Sum_{k>=0} x^(2k^2+2k)).
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EXAMPLE
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q - 2*q^3 + 3*q^9 - 2*q^11 + 2*q^17 - 2*q^19 + q^25 - 4*q^27 +...
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PROGRAM
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(PARI) {a(n)= if(n<0, 0, (-1)^n*sumdiv(2*n+1, d, (-1)^(d%8>3)))}
(PARI) {a(n)= if(n<0, 0, n=2*n+1; qfrep([1, 0; 0, 8], n)[n] -qfrep([3, 1; 1, 3], n)[n])}
(PARI) {a(n)= local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)^2*eta(x^8+A)^2/ eta(x^2+A)/eta(x^4+A), n))}
(PARI) {a(n) = if( n<0, 0, n = 2*n + 1; sumdiv(n, d, kronecker(2, d) * kronecker(-4, n/d)))}
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CROSSREFS
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A113411(n)*(-1)^n=a(n).
Sequence in context: A131636 A077888 A113411 this_sequence A099026 A053202 A050186
Adjacent sequences: A125092 A125093 A125094 this_sequence A125096 A125097 A125098
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KEYWORD
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sign
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AUTHOR
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Michael Somos, Nov 20 2006
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