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Search: id:A113417
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| A113417 |
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Expansion of q^(-1)(eta(q^2)^2*eta(q^16))^2(eta(q^4)/eta(q^8))^3 in powers of q^2. |
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+0
3
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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, -5, -2, -2, -2, -5, -2, -4, ...].
a(n)=b(2n+1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (p^(e+1)-1)/(p-1) if p == 1, 7 (mod 8), b(p^e) = ((-p)^(e+1)-1)/(-p-1) if p == 3, 5 (mod 8).
G.f.: Sum_{k>=0} a(k)x^(2k+1) = Sum_{k>0} (2k-1)*(-1)^[k/2]*x^(2k-1)/(1-x^(4k-2)) = x(Product_{k> 0} ((1-x^(2k))(1-x^(4k))(1+x^(8k)))^2/(1+x^(4k))).
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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, sumdiv(2*n+1, d, d*(d%2)*(-1)^((d+1)\4)))
(PARI) {a(n)=local(A, p, e); 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, p*=kronecker(2, p); (p^(e+1)-1)/(p-1)))))}
(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x+A)*eta(x^8+A))^2*(eta(x^2+A)/eta(x^4+A))^3, n))}
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CROSSREFS
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Sequence in context: A065075 A001370 A039794 this_sequence A113419 A133007 A141532
Adjacent sequences: A113414 A113415 A113416 this_sequence A113418 A113419 A113420
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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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