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A123330
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Expansion of eta(q^2)eta(q^3)^6/(eta(q)^2*eta(q^6)^3) in powers of q.
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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; 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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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| Expansion of c(q)^2/(3c(q^2)) in powers of q where c(q) is a cubic AGM function.
Expansion of phi(-x^3)^3/phi(-x) where phi() is a Ramanujan theta function.
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 6 sequence [ 2, 1, -4, 1, 2, -2, ...].
Moebius transform is period 6 sequence [ 2, 2, 0, -2, -2, 0, ...].
G.f.: Product_{k>0} (1+x^k)/(1-x^k)*((1-x^(3k))/(1+x^(3k)))^3.
G.f.: 1 + 2 Sum_{k>0} q^k/(1-q^k+q^(2k)) = theta_3(-q^3)^3/theta_3(-q).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v * (u - v)^2 - 2 * u * w * (v - w). - Michael Somos Aug 11 2009
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (16/3)^(1/2) (t/i) g(t) where q = exp(2 pi i t) and g() is g.f. for A107760. - Michael Somos Aug 11 2009
a(4*n) = a(3*n) = a(n). a(12*n + 10) = a(6*n + 5) = 0. - Michael Somos Aug 11 2009
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EXAMPLE
| 1 + 2*q + 4*q^2 + 2*q^3 + 2*q^4 + 4*q^6 + 4*q^7 + 4*q^8 + 2*q^9 + ... - Michael Somos Aug 11 2009
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PROG
| (PARI) {a(n)=if(n<1, n==0, 2*sumdiv(n, d, -(-1)^d*kronecker(-3, d)))}
(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)*eta(x^3+A)^6/ eta(x+A)^2/eta(x^6+A)^3, n))}
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CROSSREFS
| Cf. a(n)=2*A123331(n) if n>0. A113973(n)=(-1)^n*a(n).
2 * A033762(n) = a(2*n + 1). 2 * A033687(n) = a(3*n + 1). 2 * A112604(n) = a(4*n + 1). 2 * A112605(n) = a(4*n + 3). 2 * A097195(n) = a(6*n + 1). 2 * A123884(n) = a(12*n + 1). 4 * A121361(n) = a(12*n + 7). - Michael Somos Aug 11 2009
Sequence in context: A077748 A152753 A113973 * A194564 A072865 A179686
Adjacent sequences: A123327 A123328 A123329 * A123331 A123332 A123333
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KEYWORD
| nonn
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AUTHOR
| Michael Somos, Sep 26 2006
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