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A125061
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Expansion of psi(q)*psi(q^2)*chi(q^3)*chi(-q^6) in powers of q where psi(),chi() are Ramanujan theta functions.
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3
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1, 1, 1, 3, 1, 2, 3, 0, 1, 1, 2, 0, 3, 2, 0, 6, 1, 2, 1, 0, 2, 0, 0, 0, 3, 3, 2, 3, 0, 2, 6, 0, 1, 0, 2, 0, 1, 2, 0, 6, 2, 2, 0, 0, 0, 2, 0, 0, 3, 1, 3, 6, 2, 2, 3, 0, 0, 0, 2, 0, 6, 2, 0, 0, 1, 4, 0, 0, 2, 0, 0, 0, 1, 2, 2, 9, 0, 0, 6, 0, 2, 1, 2, 0, 0, 4, 0, 6, 0, 2, 2, 0, 0, 0, 0, 0, 3, 2, 1, 0, 3, 2, 6, 0, 2
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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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
| N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 82, Eq. (32.53).
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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 eta(q^2)*eta(q^4)^2*eta(q^6)^3/ (eta(q)*eta(q^3)*eta(q^12)^2) in powers of q.
Expansion of (theta_3(q)^2 +3*theta_3(q^3)^2)/4 in powers of q.
Euler transform of period 12 sequence [ 1, 0, 2, -2, 1, -2, 1, -2, 2, 0, 1, -2, ...].
Moebius transform is period 12 sequence [ 1, 0, 2, 0, 1, 0, -1, 0, -2, 0, -1, 0, ...].
a(n) is multiplicative with a(2^e) = 1, a(3^e) = 2-(-1)^e, a(p^e) = e+1 if p == 1 (mod 4), a(p^e) == (1-(-1)^e)/2 if p == 3 (mod 4).
G.f.: 1 +Sum_{k>0} (x^k +x^(3k))/(1 -x^(2k) +x^(4k)).
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 3 (t/i) g(t) where q = exp(2 pi i t) and g() is g.f. for A122857.
a(12*n+7) = a(12*n+11) = 0.
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PROG
| (PARI) {a(n)=if(n<1, n==0, sumdiv(n, d, ((d%2)*((d%3==0)+1))*(-1)^(d\6)))}
(PARI) {a(n)=local(A, p, e); if(n<1, n==0, A=factor(n); prod( k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, 1, if(p==3, 1+e%2*2, if(p%4==1, e+1, !(e%2)))))))}
(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)*eta(x^4+A)^2*eta(x^6+A)^3/ (eta(x+A)*eta(x^3+A)*eta(x^12+A)^2), n))}
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CROSSREFS
| Sequence in context: A057056 A016469 * A138745 A138746 A138950 A138952
Adjacent sequences: A125058 A125059 A125060 * A125062 A125063 A125064
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KEYWORD
| nonn,mult
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AUTHOR
| Michael Somos, Nov 18 2006
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