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%I #21 Nov 23 2023 08:01:16
%S 1,-2,1,1,0,-2,2,-2,1,0,0,1,2,-4,0,1,0,-2,2,0,2,0,0,-2,1,-4,1,2,0,0,2,
%T -2,0,0,0,1,2,-4,2,0,0,-4,2,0,0,0,0,1,3,-2,0,2,0,-2,0,-4,2,0,0,0,2,-4,
%U 2,1,0,0,2,0,0,0,0,-2,2,-4,1,2,0,-4,2,0,1
%N Expansion of (1 - phi(-q)^3 / phi(-q^3)) / 6 in powers of q where phi() is a Ramanujan theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%D Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 84, Eq. (32.64).
%H G. C. Greubel, <a href="/A122860/b122860.txt">Table of n, a(n) for n = 1..1000</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F Expansion of (1 + a(q) - 2*a(q^2)) / 6 = (1 - b(q)^2 / b(q^2)) / 6 in powers of q where a(), b() are cubic AGM theta functions.
%F Expansion of (1 - eta(q)^6 * eta(q^6) / (eta(q^2)^3 * eta(q^3)^2)) / 6 in powers of q.
%F Moebius transform is period 6 sequence [ 1, -3, 0, 3, -1, 0, ...].
%F a(n) is multiplicative and a(2^e) = (3(-1)^e-1)/2, a(3^e) = 1, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6).
%F a(3*n) = a(4*n) = a(n). a(6*n + 5) = 0.
%F G.f.: (1 - Product_{k>0} (1 + x^(3k)) / (1 + x^k)^3 * (1 - x^k)^3 / (1 - x^(3*k))) / 6 = Sum_{k>0} -(-x)^k / (1 + x^k + x^(2*k)).
%F G.f.: Sum_{k>0} x^(3*k-2) / (1 + x^(3*k-2)) - x^(3*k-1) / (1 + x^(3*k-1)).
%F -6 * a(n) = A122859(n) unless n=0. -(-1)^n * a(n) = A113661(n).
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 0. - _Amiram Eldar_, Nov 23 2023
%e G.f. = q - 2*q^2 + q^3 + q^4 - 2*q^6 + 2*q^7 - 2*q^8 + q^9 + q^12 + 2*q^13 + ...
%t a[ n_] := If[ n < 1, 0, -DivisorSum[ n, (-1)^(n/#) JacobiSymbol[ -3, #] &]]; (* _Michael Somos_, Feb 19 2015 *)
%o (PARI) {a(n) = if( n<1, 0, -sumdiv(n, d, (-1)^(n/d) * kronecker(-3, d)))};
%o (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, (2+(-1)^d) * kronecker(-3, d)))};
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 - eta(x + A)^6 * eta(x^6 + A) / (eta(x^2 + A)^3 * eta(x^3 + A)^2)) / 6, n))};
%o (PARI) {a(n) = if( n<1, 0, direuler(p=2, n, if(p==2, (1 - 2*X) / (1 - X^2), 1 / ((1-X) * (1 - kronecker(-3, p)*X))))[n])};
%o (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==2, (3*(-1)^e - 1) / 2, p==3, 1, p%6==1, e+1, 1-e%2 )))};
%Y Cf. A113661, A122859.
%Y Cf. A000122, A000700, A004016, A005882, A005928, A010054, A121373.
%K sign,easy,mult
%O 1,2
%A _Michael Somos_, Sep 15 2006
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