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A008442
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Expansion of Jacobi theta constant (theta_2(2z))^2/4.
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8
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1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0
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OFFSET
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1,5
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COMMENTS
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REFERENCES
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J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.26).
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LINKS
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FORMULA
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Fine gives an explicit formula for a(n) in terms of the divisors of n.
a(n) = number of divisors of n of form 8n+1, 8n+5, 8n+6 minus number of divisors of form 8n+2, 8n+3, 8n+7. [I think Fine's version is simpler - N. J. A. Sloane]
G.f.: s(8)^4/(s(4)^2), where s(k) := subs(q=q^k, eta(q)), where eta(q) is Dedekind's function, cf. A010815. [Fine]
Expansion of q * psi(q^4)^2 in powers of q where psi() is a Ramanujan theta function. - Michael Somos, Feb 22 2015
Expansion of eta(q^8)^4 / eta(q^4)^2 in powers of q.
Euler transform of period 8 sequence [ 0, 0, 0, 2, 0, 0, 0, -2, ...]. - Michael Somos, Apr 24 2004
a(n)=0 unless n=4k+1 in which case a(n) is the difference between number of divisors of n of form 4k+1 and 4k+3.
Multiplicative with a(2^e) = 0^e, a(p^e) = (1 + (-1)^e)/2 if p==3 mod 4 otherwise a(p^e) = 1+e. - Michael Somos, Sep 18 2004
Moebius transform is period 8 sequence [ 1, -1, -1, 0, 1, 1, -1, 0, ...]. - Michael Somos, Sep 02 2005
G.f.: Sum_{k>0} Kronecker(-4, k) * x^k / (1 - x^(2*k)) = Sum_{k>0} x^(2*k - 1) / (1 + x^(4*k - 2)). - Michael Somos, Sep 20 2005
G.f.: Sum_{k>0} x^k * (1 - x^k) * (1 - x^(2*k)) * (1 - x^(3*k)) / (1 - x^(8*k)) = x Product_{k>0} (1 - x^(8*k))^4 / (1 - x^(4*k))^2. - Michael Somos, Apr 24 2004
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/8 = 0.392699... (A019675). - Amiram Eldar, Oct 23 2022
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EXAMPLE
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G.f. = q + 2*q^5 + q^9 + 2*q^13 + 2*q^17 + 3*q^25 + 2*q^29 + 2*q^37 + ...
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MATHEMATICA
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a[n_] := Sum[{0, 1, -1, -1, 0, 1, 1, -1}[[Mod[d, 8] + 1]], {d, Divisors[n]}]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, May 15 2013, after Michael Somos *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^2]^2 / 4, {x, 0, n}]; (* Michael Somos, Feb 22 2015 *)
a[ n_] := If[ n < 1 || Mod[n, 4] != 1, 0, Sum[ KroneckerSymbol[ 4, d], {d, Divisors @n}]]; (* Michael Somos, Feb 22 2015 *)
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PROG
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(PARI) {a(n) = if( n<1 || n%4!=1, 0, sumdiv(n, d, (d%4==1) - (d%4==3)))}; /* Michael Somos, Apr 24 2004 */
(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, [0, 1, -1, -1, 0, 1, 1, -1][d%8+1]))}; /* Michael Somos, Apr 24 2004 */
(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^8 + A)^4 / eta(x^4 + A)^2, n))}; /* Michael Somos, Apr 24 2004 */
(Magma) A := Basis( ModularForms( Gamma1(16), 1), 106); A[2] + 2*A[6]; /* Michael Somos, Feb 22 2015 */
(Python)
from sympy import divisors
def A008442(n): return 0 if n&3!=1 else sum(((a:=d&3)==1)-(a==3) for d in divisors(n, generator=True)) # Chai Wah Wu, May 17 2023
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CROSSREFS
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KEYWORD
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nonn,mult
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AUTHOR
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STATUS
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approved
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