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A132003
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Expansion of phi(q^3)/ phi(q) *phi(-q^2)* phi(-q^6) in powers of q where phi() is a Ramanujan theta function.
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2
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1, -2, 2, -2, 2, -4, 2, 0, 2, -2, 4, 0, 2, -4, 0, -4, 2, -4, 2, 0, 4, 0, 0, 0, 2, -6, 4, -2, 0, -4, 4, 0, 2, 0, 4, 0, 2, -4, 0, -4, 4, -4, 0, 0, 0, -4, 0, 0, 2, -2, 6, -4, 4, -4, 2, 0, 0, 0, 4, 0, 4, -4, 0, 0, 2, -8, 0, 0, 4, 0, 0, 0, 2, -4, 4, -6, 0, 0, 4, 0, 4, -2, 4, 0, 0, -8, 0, -4, 0, -4, 4, 0, 0, 0, 0, 0, 2, -4, 2, 0, 6, -4, 4, 0, 4
(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
| N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.72).
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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)* eta(q^6)^7/( eta(q^2)^3* eta(q^3)^2* eta(q^12)^3) in powers of q.
a(n)= -2*b(n) where b(n) is multiplicative with b(2^e) = 2*0^e -1, b(3^e) = 1, b(p^e) = e+1 if p == 1 (mod 4), b(p^e) = (1+(-1)^e)/2 if p == 3 (mod 4).
Euler transform of period 12 sequence [ -2, 1, 0, 0, -2, -4, -2, 0, 0, 1, -2, -2, ...].
G.f.: 1 -2* Sum_{k>0} x^k/(x^k+1)* kronecker(-36, k).
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PROG
| (PARI) {a(n)= if(n<1, n==0, -2*sumdiv(n, d, (-1)^(n+d)* kronecker(-36, d)))}
(PARI) {a(n)= local(A); if(n<0, 0, A= x*O(x^n); polcoeff( eta(x+A)^2* eta(x^4+A)* eta(x^6+A)^7/( eta(x^2+A)^3* eta(x^3+A)^2* eta(x^12+A)^3), n))}
(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==3, 1, if(p==2, -1, if(p%4==1, e+1, !(e%2)))))))}
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CROSSREFS
| -2*A132004(n)= a(n) unless n=0.
Sequence in context: A092904 A062816 A122857 * A109810 A122066 A053238
Adjacent sequences: A132000 A132001 A132002 * A132004 A132005 A132006
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Aug 06 2007
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