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A035172
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a(n) = Sum_{d|n} Kronecker(-18, d).
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2
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1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 2, 1, 0, 0, 0, 1, 2, 1, 2, 0, 0, 2, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 2, 2, 0, 1, 0, 2, 0, 0, 2, 0, 2, 2, 0, 0, 0, 1, 1, 1, 2, 0, 0, 1, 0, 0, 2, 0, 2, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 0, 0, 1, 2, 0, 1, 2, 0, 0, 0, 0, 1, 2, 2, 0, 0, 2, 0, 2, 2, 0, 0, 0, 0, 0, 0, 1, 2, 1, 2, 1, 0, 2, 0, 0, 0
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OFFSET
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1,11
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COMMENTS
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For n nonzero, a(n) is nonzero if and only if n is in A002479.
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REFERENCES
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Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 81, Eq. (32.51).
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LINKS
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FORMULA
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Expansion of q * psi(-q^3) * psi(-q^6) * chi(-q^4) / chi(-q) in powers of q where psi(), chi() are Ramanujan theta functions.
G.f.: x * Product_{k>0} (1 - x^(3*k)) * (1 - x^(24*k)) * (1 + x^k) / (1 + x^(4*k)).
Euler transform of period 24 sequence [ 1, 0, 0, -1, 1, -1, 1, 0, 0, 0, 1, -2, 1, 0, 0, 0, 1, -1, 1, -1, 0, 0, 1, -2, ...]. (End)
Moebius transform is period 24 sequence [ 1, 0, 0, 0, -1, 0, -1 ,0, 0, 0, 1, 0, -1, 0, 0, 0, 1, 0, 1, 0, 0, 0, -1, 0, ...]. - Michael Somos, Jan 28 2006
Multiplicative with a(2^e) = a(3^e) = 1, a(p^e) = e+1 if p == 1,3 (mod 8), a(p^e) = (1 + (-1)^e)/2 if p == 5,7 (mod 8).
G.f.: Sum_{k>0} x^k * (1 - x^(4*k)) * (1 - x^(6* k)) / (1 + x^(12*k)). (End)
Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = -18.
G.f.: 1 + Sum{n = -infinity...infinity} (q^n - q^(5n)) / (1 + q^(12n)) (see Berkovich/Yesilyurt). - Ralf Stephan, May 14 2007
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(3*sqrt(2)) = 0.7404804... (A093825). - Amiram Eldar, Nov 16 2023
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MATHEMATICA
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PROG
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(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, kronecker( -18, d)))}
(PARI) {a(n) = if( n<1, 0, direuler( p=2, n, 1 / (1 - X) / (1 - kronecker( -18, p) * X))[n])}
(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^24 + A) / eta(x + A) / eta(x^8 + A), n))}
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CROSSREFS
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KEYWORD
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nonn,easy,mult
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AUTHOR
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STATUS
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approved
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