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Search: id:A115660
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| A115660 |
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Expansion of (phi(q)phi(q^6)-phi(q^2)phi(q^3))/2 where phi() is a Ramanujan theta function. |
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3
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| 1, -1, -1, 1, -2, 1, 2, -1, 1, 2, -2, -1, 0, -2, 2, 1, 0, -1, 0, -2, -2, 2, 0, 1, 3, 0, -1, 2, -2, -2, 2, -1, 2, 0, -4, 1, 0, 0, 0, 2, 0, 2, 0, -2, -2, 0, 0, -1, 3, -3, 0, 0, -2, 1, 4, -2, 0, 2, -2, 2, 0, -2, 2, 1, 0, -2, 0, 0, 0, 4, 0, -1, 2, 0, -3, 0, -4, 0, 2, -2, 1, 0, -2, -2, 0, 0, 2, 2, 0, 2, 0, 0, -2, 0, 0, 1, 2, -3, -2, 3, -2, 0, 2, 0, 4
(list; graph; listen)
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OFFSET
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1,5
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LINKS
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A. Berkovich and H. Yesilyurt, Ramanujan's identities and representation of integers by certain binary and quaternary quadratic forms
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FORMULA
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Expansion of eta(q)eta(q^4)eta(q^6)eta(q^24)/(eta(q^3)eta(q^8)) in powers of q.
Euler transform of period 24 sequence [ -1, -1, 0, -2, -1, -1, -1, -1, 0, -1, -1, -2, -1, -1, 0, -1, -1, -1, -1, -2, 0, -1, -1, -2, ...].
Multiplicative with a(2^e) = a(3^e) = (-1)^e, a(p^e) = e+1 if p == 1, 7 (mod 24), a(p^e) = (e+1)(-1)^e if p == 5, 11 (mod 24), a(p^e) = (1+(-1)^e)/2 if p == 13, 17, 19, 23 (mod 24).
G.f.: Sum_{k>0} kronecker(k,8)*x^k/(1+x^k+x^(2k)) = Sum_{k>0} kronecker(k,3)*x^k(1-x^(2k))/(1+x^(4k)).
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EXAMPLE
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q -q^2 -q^3 +q^4 -2*q^5 +q^6 +2*q^7 -q^8 +q^9 +2*q^10 +...
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PROGRAM
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(PARI) {a(n)=local(A, p, e); 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|p==3, (-1)^e, if(p%24<12, (e+1)*kronecker(-12, p)^e, (1+(-1)^e)/2)))))}
(PARI) {a(n)=local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff( eta(x+A)*eta(x^4+A)*eta(x^6+A)*eta(x^24+A)/(eta(x^3+A)*eta(x^8+A)), n))}
(PARI) a(n)=if(n<1, 0, sumdiv(n, d, kronecker(d, 8)*kronecker(n/d, 3)))
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CROSSREFS
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Cf. A000377(n)=|a(n)|.
Sequence in context: A075117 A029810 A000377 this_sequence A128581 A026517 A072047
Adjacent sequences: A115657 A115658 A115659 this_sequence A115661 A115662 A115663
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KEYWORD
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sign,mult
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AUTHOR
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Michael Somos, Jan 28 2006
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