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Search: id:A106406
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| A106406 |
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Expansion of (eta(q)eta(q^15))^2/(eta(q^3)eta(q^5)) in powers of q. |
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| 1, -2, -1, 3, -1, 2, 0, -4, 1, 2, 0, -3, 0, 0, 1, 5, -2, -2, 2, -3, 0, 0, -2, 4, 1, 0, -1, 0, 0, -2, 2, -6, 0, 4, 0, 3, 0, -4, 0, 4, 0, 0, 0, 0, -1, 4, -2, -5, 1, -2, 2, 0, -2, 2, 0, 0, -2, 0, 0, 3, 2, -4, 0, 7, 0, 0, 0, -6, 2, 0, 0, -4, 0, 0, -1, 6, 0, 0, 2, -5, 1, 0, -2, 0, 2, 0, 0, 0, 0, 2, 0, -6, -2, 4, -2, 6, 0, -2, 0, 3, 0, -4, 0, 0, 0
(list; graph; listen)
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OFFSET
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1,2
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FORMULA
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Euler transform of period 15 sequence [ -2, -2, -1, -2, -1, -1, -2, -2, -1, -1, -2, -1, -2, -2, -2, ...].
G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w)=-v^3+4uvw+2uw^2+u^2w.
G.f.: Sum_{k>0} kronecker(k, 3) x^k(1-x^k)(1-x^(2k))/(1-x^(5k)) = Sum_{k>0} kronecker(k, 5) x^k(1-x^k)/(1-x^(3k)).
a(n) is multiplicative with a(3^e) = a(5^e) = (-1)^e, a(p^e) = (1+(-1)^e)/2 if p == 7, 11, 13, 14 (mod 15), a(p^e) = e+1 if p == 1, 4 (mod 15), a(p^e) = (e+1)(-1)^e if p == 2, 8 (mod 15). - Michael Somos Oct 19 2005
a(15n+7)=a(15n+11)=a(15n+13)=a(15n+14)=0.
G.f.: (1/2)(Sum_{n,m} x^(n^2+nm+4m^2) -x^(2n^2+nm+2m^2)). - Michael Somos Aug 25 2006
G.f.: x*Product_{k>0} ((1-x^k)(1-x^(15k)))^2/((1-x^(3k))(1-x^(5k))).
A035175(n)=|a(n)|. a(n)>0 iff n in A028957. a(n)<0 iff n in A028955.
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EXAMPLE
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q - 2*q^2 - q^3 + 3*q^4 - q^5 + 2*q^6 - 4*q^8 + q^9 + 2*q^10 +...
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PROGRAM
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(PARI) {a(n)=local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff( eta(x+A)^2*eta(x^15+A)^2/(eta(x^3+A)*eta(x^5+A)), n))}
(PARI) a(n)=if(n<1, 0, sumdiv(n, d, kronecker(d, 3)*kronecker(n/d, 5)))
(PARI) {a(n)=local(A, p, e, x); if(n<1, 0, A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==3|p==5, (-1)^e, if((p%15)!=2^(x=valuation(p%15, 2)), (e+1)%2, (e+1)*(-1)^(x*e))))))}
(PARI) {a(n)=if(n<1, 0, (qfrep([2, 1; 1, 8], n, 1)-qfrep([4, 1; 1, 4], n, 1))[n])} /* Michael Somos Aug 25 2006 */
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CROSSREFS
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Sequence in context: A156248 A123864 A035175 this_sequence A092412 A078734 A028293
Adjacent sequences: A106403 A106404 A106405 this_sequence A106407 A106408 A106409
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KEYWORD
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sign,mult
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AUTHOR
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Michael Somos, May 02 2005
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