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Search: id:A113419
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| A113419 |
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Expansion of q^(-1)(eta(q^4)^9*eta(q^16)^2)/(eta(q^2)^2*eta(q^8)^5) in powers of q^2. |
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+0
1
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| 1, 2, -4, -8, 7, 10, -12, -8, 18, 18, -16, -24, 21, 20, -28, -32, 20, 32, -36, -24, 42, 42, -28, -48, 57, 36, -52, -40, 36, 58, -60, -56, 48, 66, -48, -72, 74, 42, -80, -80, 61, 82, -72, -56, 90, 96, -64, -72, 98, 70, -100, -104, 64, 106, -108, -72, 114, 96, -84, -144, 111, 84, -104, -128, 84, 130, -144
(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 8 sequence [2, -7, 2, -2, 2, -7, 2, -4, ...].
a(n)=b(2n+1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (x^(e+1)-y^(e+1))/(x-y) where x=p*(-1)^[p/4] and y = (-1)^[p/2].
G.f.: Sum_{k>0} (2k-1)*(-1)^[(k-1)/2]*x^(2k-1)/(1+x^(4k-2)).
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EXAMPLE
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q +2*q^3 -4*q^5 -8*q^7 +7*q^9 +10*q^11 -12*q^13 -8*q^15 +...
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PROGRAM
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(PARI) a(n)=if(n<0, 0, n=2*n+1; sumdiv(n, d, d*(d%2)*(-1)^((n/d)\2+(d-1)\4)))
(PARI) {a(n)=local(A, p, e, t); if(n<0, 0, n=2*n+1; A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, 0, t=(-1)^(p\2); p*=kronecker(-2, p); (p^(e+1)-t^(e+1))/(p-t)))))}
(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^9*eta(x^8+A)^2/(eta(x+A)^2*eta(x^4+A)^5), n))}
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CROSSREFS
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A113417(n) = (-1)^n*a(n).
Sequence in context: A001370 A039794 A113417 this_sequence A133007 A141532 A083550
Adjacent sequences: A113416 A113417 A113418 this_sequence A113420 A113421 A113422
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KEYWORD
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sign
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AUTHOR
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Michael Somos, Oct 29 2005
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