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Search: id:A113421
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| A113421 |
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Expansion of eta(q)^2*eta(q^4)*eta(q^6)^2*eta(q^12)/eta(q^3)^2 in powers of q. |
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+0
1
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| 1, -2, -1, 4, -4, 2, 6, -8, 1, 8, -12, -4, 14, -12, 4, 16, -16, -2, 18, -16, -6, 24, -24, 8, 21, -28, -1, 24, -28, -8, 30, -32, 12, 32, -24, 4, 38, -36, -14, 32, -40, 12, 42, -48, -4, 48, -48, -16, 43, -42, 16, 56, -52, 2, 48, -48, -18, 56, -60, 16, 62, -60, 6, 64, -56, -24, 66, -64, 24, 48, -72, -8, 74, -76, -21, 72
(list; graph; listen)
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OFFSET
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1,2
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FORMULA
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Euler transform of period 12 sequence [ -2, -2, 0, -3, -2, -2, -2, -3, 0, -2, -2, -4, ...].
G.f.: Sum_{k>0} (3k-2)x^(3k-2)/(1+x^(6k-4)) - (3k-1)x^(3k-1)/(1+x^(6k-2)) = Sum_{k>0} -(-1)^k* x^(2k-1)*(1-x^(2k-1))^2*(1-x^(4k-2))/(1-x^(6k-3))^2.
a(n) is multiplicative and a(2^e) = (-2)^e, a(3^e) = (-1)^e, a(p^e) = (x^(e+1)-y^(e+1))/(x-y) where x=p*kronecker(-3, p) and y=(-1)^[p/2].
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EXAMPLE
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q -2*q^2 -q^3 +4*q^4 -4*q^5 +2*q^6 +6*q^7 -8*q^8 +q^9 +8*q^10 +...
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PROGRAM
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(PARI) a(n)=if(n<1, 0, sumdiv(n, d, (n/d%2)*d*kronecker(-3, d)*(-1)^(n/d\2)))
(PARI) {a(n)=local(A, p, e, t); if(n<1, 0, A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, (-2)^e, if(p==3, (-1)^e, t=(-1)^(p\2); p*=kronecker(-3, p); (p^(e+1)-t^(e+1))/(p-t))))))}
(PARI) {a(n)=local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff( eta(x+A)^2*eta(x^4+A)*eta(x^6+A)^2*eta(x^12+A)/eta(x^3+A)^2, n))}
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CROSSREFS
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Adjacent sequences: A113418 A113419 A113420 this_sequence A113422 A113423 A113424
Sequence in context: A107728 A128250 A086145 this_sequence A135366 A051289 A090802
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KEYWORD
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sign,mult
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AUTHOR
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Michael Somos, Oct 29 2005
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