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A115660
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Expansion of (phi(q)phi(q^6)-phi(q^2)phi(q^3))/2 where phi() is a Ramanujan theta function.
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3
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1, -1, -1, 1, -2, 1, 2, -1, 1, 2, -2, -1, 0, -2, 2, 1, 0, -1, 0, -2, -2, 2, 0, 1, 3, 0, -1, 2, -2, -2, 2, -1, 2, 0, -4, 1, 0, 0, 0, 2, 0, 2, 0, -2, -2, 0, 0, -1, 3, -3, 0, 0, -2, 1, 4, -2, 0, 2, -2, 2, 0, -2, 2, 1, 0, -2, 0, 0, 0, 4, 0, -1, 2, 0, -3, 0, -4, 0, 2, -2, 1, 0, -2, -2, 0, 0, 2, 2, 0, 2, 0, 0, -2, 0, 0, 1, 2, -3, -2, 3, -2, 0, 2, 0, 4
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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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
A. Berkovich and H. Yesilyurt, Ramanujan's identities and representation of integers by certain binary and quaternary quadratic forms
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FORMULA
| Expansion of eta(q)eta(q^4)eta(q^6)eta(q^24)/(eta(q^3)eta(q^8)) in powers of q.
Euler transform of period 24 sequence [ -1, -1, 0, -2, -1, -1, -1, -1, 0, -1, -1, -2, -1, -1, 0, -1, -1, -1, -1, -2, 0, -1, -1, -2, ...].
Multiplicative with a(2^e) = a(3^e) = (-1)^e, a(p^e) = e+1 if p == 1, 7 (mod 24), a(p^e) = (e+1)(-1)^e if p == 5, 11 (mod 24), a(p^e) = (1+(-1)^e)/2 if p == 13, 17, 19, 23 (mod 24).
G.f.: Sum_{k>0} kronecker(k,8)*x^k/(1+x^k+x^(2k)) = Sum_{k>0} kronecker(k,3)*x^k(1-x^(2k))/(1+x^(4k)).
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EXAMPLE
| q -q^2 -q^3 +q^4 -2*q^5 +q^6 +2*q^7 -q^8 +q^9 +2*q^10 +...
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PROG
| (PARI) {a(n)=local(A, p, e); if(n<1, 0, A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2|p==3, (-1)^e, if(p%24<12, (e+1)*kronecker(-12, p)^e, (1+(-1)^e)/2)))))}
(PARI) {a(n)=local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff( eta(x+A)*eta(x^4+A)*eta(x^6+A)*eta(x^24+A)/(eta(x^3+A)*eta(x^8+A)), n))}
(PARI) a(n)=if(n<1, 0, sumdiv(n, d, kronecker(d, 8)*kronecker(n/d, 3)))
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CROSSREFS
| Cf. A000377(n)=|a(n)|.
Sequence in context: A190611 * A128581 A192013 A026517 A072047 A106802
Adjacent sequences: A115657 A115658 A115659 * A115661 A115662 A115663
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KEYWORD
| sign,mult
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AUTHOR
| Michael Somos, Jan 28 2006
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