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Search: id:A113661
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| A113661 |
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Expansion of (phi(x)^3/phi(x^3)-1)/6 where phi() is a Ramanujan theta function. |
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+0
3
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| 1, 2, 1, -1, 0, 2, 2, 2, 1, 0, 0, -1, 2, 4, 0, -1, 0, 2, 2, 0, 2, 0, 0, 2, 1, 4, 1, -2, 0, 0, 2, 2, 0, 0, 0, -1, 2, 4, 2, 0, 0, 4, 2, 0, 0, 0, 0, -1, 3, 2, 0, -2, 0, 2, 0, 4, 2, 0, 0, 0, 2, 4, 2, -1, 0, 0, 2, 0, 0, 0, 0, 2, 2, 4, 1, -2, 0, 4, 2, 0, 1, 0, 0, -2, 0, 4, 0, 0, 0, 0, 4, 0, 2, 0, 0, 2, 2, 6, 0, -1, 0, 0, 2, 4, 0
(list; graph; listen)
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OFFSET
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1,2
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REFERENCES
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B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 227 Entry 4(iv).
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FORMULA
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a(n) is multiplicative and a(2^e) = (1-3(-1)^e)/2 if e>0, a(3^e) = 1, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6).
Moebius transform is period 12 sequence [1, 1, 0, -3, -1, 0, 1, 3, 0, -1, -1, 0, ...].
Expansion of ((eta(q^2)^15*eta(q^3)^2*eta(q^12)^2)/(eta(q)^6*eta(q^4)^6*eta(q^6)^5)-1)/6 in powers of.
G.f.: Sum_{k>0} x^k/(1+x^k+x^(2k)) +2*x^(4k-2)/(1+x^(4k-2)+x^(8k-4)).
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PROGRAM
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(PARI) {a(n)=local(x); if(n<1, 0, x=valuation(n, 2); if(n%2, 1, (1-3*(-1)^x)/2)*sumdiv(n/2^x, d, kronecker(-3, d)))}
(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, (1-3*(-1)^e)/2, if(p==3, 1, if(p%6==1, e+1, !(e%2)))))))}
(PARI) {a(n)=if(n<1, 0, direuler(p=2, n, if(p==2, 2-(1-2*X)/(1-X^2), 1/(1-X)/(1-kronecker(-3, p)*X)))[n])}
(PARI) {a(n)=local(A); if(n<1, 0, A=x*O(x^n); polcoeff( (eta(x^2+A)^15*eta(x^3+A)^2*eta(x^12+A)^2/ eta(x+A)^6/eta(x^4+A)^6/eta(x^6+A)^5-1)/6, n))}
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CROSSREFS
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Cf. A113660(n)=6*a(n), if n>0.
Adjacent sequences: A113658 A113659 A113660 this_sequence A113662 A113663 A113664
Sequence in context: A060450 A025860 A058394 this_sequence A113974 A122860 A123331
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KEYWORD
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sign,mult
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AUTHOR
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Michael Somos, Nov 03 2005
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