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Search: id:A122274
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| A122274 |
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Expansion of q^(-1/3)*eta(q)^3*eta(q^5) in powers of q. |
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+0
2
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| 1, -3, 0, 5, 0, -1, -4, 0, -5, 0, 8, 10, 0, -5, 0, -20, 7, 0, 0, 0, 2, 13, 0, 0, 0, 12, -16, 0, -10, 0, 0, -20, 0, 15, 0, 10, 14, 0, 20, 0, -11, -24, 0, 0, 0, -10, -4, 0, 0, 0, 8, 0, 0, -15, 0, 40, 13, 0, 0, 0, -22, 20, 0, -20, 0, -21, -16, 0, 0, 0, -28, 40, 0, 0, 0, 10, 26, 0, 20, 0, 2, -6, 0, -25, 0, -9, 0, 0, -10
(list; graph; listen)
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OFFSET
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0,2
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FORMULA
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Euler transform of period 5 sequence [ -3, -3, -3, -3, -4, ...].
Define c(3k+1)=A122274(k), c(3k+2)=sqrt(-5)*A122275(k), c(3k)=0. Then c(n) is multiplicative with c(3^e)=0^e, c(p^e) = p^(e/2)*(1+(-1)^e)/2 if p == 7, 13 (mod 15), c(p^e) = (-p)^(e/2)*(1+(-1)^e)/2 if p == 11, 14 (mod 15), c(p^e) = c(p)c(p^(e-1))-p*c(p^(e-2)) if p == 1, 4 (mod 15), c(p^e) = c(p)c(p^(e-1))+p*c(p^(e-2)) if p == 2, 8 (mod 15), where c(p) is determined by c(2)=1, or c(p)=2*x when p = x^2 +15*y^2, x == 1 (mod 3) for p == 1, 4 (mod 15), and c(p)=2*x*sqrt(-5) when p = 5*x^2 +3*y^2, x == 2 (mod 3) for p == 2, 8 (mod 15).
a(5n+2)=a(5n+4)=0.
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PROGRAM
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(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)^3*eta(x^5+A), n))}
(PARI) {a(n)=local(A, p, e, w, x, y, z, a0, a1); if(n<0, 0, n=3*n+1; A=factor(n); w=quadgen(-20); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==3, 0, z=p*kronecker(5, p); if(p==2, x=1, if(p==5, x=-1, if((p%15)!=2^valuation(p%15, 2), x=0, if(valuation(p%15, 2)%2, for(y=1, sqrtint(p\3), if(issquare((p-3*y^2)/5, &x), break)), for(y=1, sqrtint(p\15), if(issquare(p-15*y^2, &x), break))); x*=2*(-1)^(p%3!=x%3)))); a0=1; a1=y=x*if(p%3==1, 1, w); for(i=2, e, x=y*a1-z*a0; a0=a1; a1=x); a1))))}
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CROSSREFS
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5*A122275(n)=a(5n+3). A122275(5n+1)=-a(n).
Sequence in context: A136599 A131986 A002656 this_sequence A003966 A123931 A058026
Adjacent sequences: A122271 A122272 A122273 this_sequence A122275 A122276 A122277
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KEYWORD
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sign
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AUTHOR
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Michael Somos, Sep 08 2006
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