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Search: id:A113973
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| A113973 |
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Expansion of phi(x^3)^3/phi(x) where phi() is a Ramanujan theta function. |
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4
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| 1, -2, 4, -2, 2, 0, 4, -4, 4, -2, 0, 0, 2, -4, 8, 0, 2, 0, 4, -4, 0, -4, 0, 0, 4, -2, 8, -2, 4, 0, 0, -4, 4, 0, 0, 0, 2, -4, 8, -4, 0, 0, 8, -4, 0, 0, 0, 0, 2, -6, 4, 0, 4, 0, 4, 0, 8, -4, 0, 0, 0, -4, 8, -4, 2, 0, 0, -4, 0, 0, 0, 0, 4, -4, 8, -2, 4, 0, 8, -4, 0, -2, 0, 0, 4, 0, 8, 0, 0, 0, 0, -8, 0, -4, 0, 0, 4, -4, 12, 0, 2, 0, 0, -4, 8
(list; graph; listen)
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OFFSET
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0,2
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REFERENCES
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B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 375 Entry 35.
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FORMULA
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a(n)=-2*b(n) where b(n) is multiplicative and b(2^e) = (1-3(-1)^e)/2 if e>0, b(3^e) = 1, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6).
Euler transform of period 12 sequence [ -2, 3, 4, 1, -2, -6, -2, 1, 4, 3, -2, -2, ...].
Moebius transform is period 12 sequence [ -2, 6, 0, -2, 2, 0, -2, 2, 0, -6, 2, 0, ...].
Expansion of (eta(q)^2*eta(q^4)^2*eta(q^6)^15)/ (eta(q^2)^5*eta(q^3)^6*eta(q^12)^6) in powers of q.
G.f.: theta_3(q^3)^3/theta_3(q).
G.f.: 1+2( Sum_{k>0} x^(3k-1)/(1-(-x)^(3k-1)) - x^(3k-2)/(1-(-x)^(3k-2))) = 1 +2( Sum_{k>0} (-1)^k x^k/(1+x^k+x^(2k)) +2 x^(4k)/(1+x^(4k)+x^(8k)) ).
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PROGRAM
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(PARI) {a(n)=local(x); if(n<1, n==0, x=valuation(n, 2); if(n%2, -2, (3-(-1)^x))*sumdiv(n/2^x, d, kronecker(-3, d)))}
(PARI) {a(n)=local(A, p, e); if(n<1, n==0, A=factor(n); -2*prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, (-3+(-1)^e)/2, if(p==3, 1, if(p%6==1, e+1, !(e%2)))))))}
(PARI) {a(n)=if(n<1, n==0, -2*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<0, 0, A=sum(k=1, sqrtint(n), 2*x^k^2, 1+x*O(x^n)); polcoeff( subst(A+x*O(x^(n\3)), x, x^3)^3/A, n))}
(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)^2*eta(x^4+A)^2*eta(x^6+A)^15/ eta(x^2+A)^5/eta(x^3+A)^6/eta(x^12+A)^6, n))}
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CROSSREFS
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a(n)=-2*A113974(n) if n>0.
Sequence in context: A105791 A116515 A037178 this_sequence A123330 A072865 A013604
Adjacent sequences: A113970 A113971 A113972 this_sequence A113974 A113975 A113976
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KEYWORD
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sign
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AUTHOR
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Michael Somos, Nov 10 2005
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