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A123477
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Expansion of (1 - b(q)) / 3 in powers of q where b(q) is a cubic AGM theta function.
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5
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1, 0, -2, 1, 0, 0, 2, 0, -2, 0, 0, -2, 2, 0, 0, 1, 0, 0, 2, 0, -4, 0, 0, 0, 1, 0, -2, 2, 0, 0, 2, 0, 0, 0, 0, -2, 2, 0, -4, 0, 0, 0, 2, 0, 0, 0, 0, -2, 3, 0, 0, 2, 0, 0, 0, 0, -4, 0, 0, 0, 2, 0, -4, 1, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, -2, 2, 0, 0, 2, 0, -2, 0, 0, -4, 0, 0, 0, 0, 0, 0, 4, 0, -4, 0, 0, 0, 2, 0, 0, 1, 0, 0, 2, 0, 0
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OFFSET
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1,3
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COMMENTS
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Denoted by lambda(n) on page 4 (1.7) in Kassel and Reutenauer arXiv:1610.07793. - Michael Somos, Dec 10 2017
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LINKS
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FORMULA
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Moebius transform is period 9 sequence [1, -1, -3, 1, -1, 3, 1, -1, 0, ...].
a(n) is multiplicative and a(p^e) = -2 if p = 3 and e>0, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1+(-1)^e)/2 if p == 2, 5 (mod 6).
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EXAMPLE
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G.f. = q - 2*q^3 + q^4 + 2*q^7 - 2*q^9 - 2*q^12 + 2*q^13 + q^16 + 2*q^19 + ...
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MAPLE
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local a, pe, p, e;
a := 1;
for pe in ifactors(n)[2] do
p := op(1, pe) ;
e := op(2, pe) ;
if modp(p, 6) = 1 then
a := a*(e+1) ;
elif modp(p, 6) in {2, 5} then
a := a*(1+(-1)^e)/2 ;
elif e > 0 then
a := -2*a ;
end if;
end do:
a ;
end proc:
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MATHEMATICA
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a[ n_] := If[ n < 1, 0, DivisorSum[ n, {1, -1, -3, 1, -1, 3, 1, -1, 0} [[Mod[#, 9, 1]]] &]]; (* Michael Somos, Dec 10 2017 *)
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PROG
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(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, [0, 1, -1, -3, 1, -1, 3, 1, -1] [d%9+1]))};
(PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, -2, p%6==1, e+1, !(e%2))))};
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CROSSREFS
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KEYWORD
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sign,mult
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AUTHOR
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STATUS
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approved
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