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A113660
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Expansion of phi(x)^3/phi(x^3) where phi() is a Ramanujan theta function.
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3
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1, 6, 12, 6, -6, 0, 12, 12, 12, 6, 0, 0, -6, 12, 24, 0, -6, 0, 12, 12, 0, 12, 0, 0, 12, 6, 24, 6, -12, 0, 0, 12, 12, 0, 0, 0, -6, 12, 24, 12, 0, 0, 24, 12, 0, 0, 0, 0, -6, 18, 12, 0, -12, 0, 12, 0, 24, 12, 0, 0, 0, 12, 24, 12, -6, 0, 0, 12, 0, 0, 0, 0, 12, 12, 24, 6, -12, 0, 24, 12, 0, 6, 0, 0, -12, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
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REFERENCES
| B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 227 Entry 4(iv).
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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| a(n)=6*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 [6, -9, 4, -3, 6, -6, 6, -3, 4, -9, 6, -2, ...].
Moebius transform is period 12 sequence [6, 6, 0, -18, -6, 0, 6, 18, 0, -6, -6, 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) in powers of.
G.f.: 1+6( 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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PROG
| (PARI) {a(n)=local(x); if(n<1, n==0, x=valuation(n, 2); if(n%2, 2, (1-3*(-1)^x))*3*sumdiv(n/2^x, d, kronecker(-3, d)))}
(PARI) {a(n)=local(A, p, e); if(n<1, n==0, A=factor(n); 6*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, n==0, 6*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=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, n))}
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CROSSREFS
| Cf. a(n)=6*A113661(n), if n>0.
Sequence in context: A064913 A066401 A076590 * A122859 A050496 A103698
Adjacent sequences: A113657 A113658 A113659 * A113661 A113662 A113663
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Nov 03 2005
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EXTENSIONS
| Corrected by Charles R Greathouse IV (charles.greathouse(AT)case.edu), Sep 02 2009
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