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Search: id:A106401
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| A106401 |
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Expansion of (eta(q)eta(q^9))^3/eta(q^3)^2 in powers of q. |
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1
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| 1, -3, 0, 7, -6, 0, 8, -15, 0, 18, -12, 0, 14, -24, 0, 31, -18, 0, 20, -42, 0, 36, -24, 0, 31, -42, 0, 56, -30, 0, 32, -63, 0, 54, -48, 0, 38, -60, 0, 90, -42, 0, 44, -84, 0, 72, -48, 0, 57, -93, 0, 98, -54, 0, 72, -120, 0, 90, -60, 0, 62, -96, 0, 127, -84, 0, 68, -126, 0, 144, -72, 0, 74, -114, 0, 140, -96, 0
(list; graph; listen)
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OFFSET
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1,2
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FORMULA
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G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w)=-v^3+6uvw+4uw^2+u^2w.
Euler transform of period 9 sequence [ -3, -3, -1, -3, -3, -1, -3, -3, -4, ...].
a(n) is multiplicative with a(3^e) = 0 if e>0, a(p^e) = (p^(e+1)-1)/(p-1) if e even or p == 1 (mod 3), a(p^e) = -(p^(e+1)-1)/(p-1) otherwise. - Michael Somos Oct 19 2005
a(3n)=0.
Expansion of b(q)*c(q^3)/3 in powers of q where b(),c() are cubic AGM analog functions. - Michael Somos Oct 17 2006
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EXAMPLE
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q - 3*q^2 + 7*q^4 - 6*q^5 + 8*q^7 - 15*q^8 + 18*q^10 - 12*q^11 +...
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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)*eta(x^9+A))^3/eta(x^3+A)^2, n))}
(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==3, 0, (-1)^((p%3>1)*e)*(p^(e+1)-1)/(p-1)))))}
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CROSSREFS
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A033686(n)=-a(3n+2)/3.
Sequence in context: A115860 A104687 A155831 this_sequence A011200 A021329 A019970
Adjacent sequences: A106398 A106399 A106400 this_sequence A106402 A106403 A106404
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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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