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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, -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, 2, 1, 0, 0, 2, 0, 0, 0, 0, -2, 2, -4, 1, 2, 0, -4, 2, 0, 1, 0, 0, 2, 0, -4, 0, 0, 0, 0, 4, 0, 2, 0, 0, -2, 2, -6, 0, 1, 0, 0, 2, -4, 0

0a[ n_] := If[ n < 1, 0, -DivisorSum[ n, (-1)^(n/#) JacobiSymbol[ -3, #] &]]; (* Michael Somos, Feb 19 2015 *)

(PARI) {a(n) = localmy(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))};

(PARI) {a(n) = localmy(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==2, (3*(-1)^e - 1) / 2, if(p==3, 1, if(p%6==1, e+1, !(1-e%2)))) )))};

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Expansion of (1 - phi(-q)^3 / phi(-q^3)) / 6 in powers of q where phi() is a Ramanujan theta function.

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

M. Somos, <a href="http://cis.csuohio.edu/~somos.crg4.com/multiq.pdfhtml">Introduction to Ramanujan theta functions</a>

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 analog theta functions.

Expansion of (1 - eta(q)^6 * eta(q^6) / (eta(q^2)^3 * eta(q^3)^2)) / 6 in powers of q.

a(3n3*n) = a(4n4*n) = a(n). a(6n6*n + 5) = 0.

G.f.: (1 - Product_{k>0} (1 + x^(3k)) / (1 + x^k)^3 * (1 - x^k)^3 / (1 - x^(3k3*k))) / 6 = Sum_{k>0} -(-x)^k / (1 + x^k + x^(2k2*k)).

G.f.: Sum_{k>0} x^(3*k-2) / (1 + x^(3*k-2)) - x^(3*k-1) / (1 + x^(3*k-1)).

-6 * a(n) = A122859(n) unless n=0. -(-1)^n * a(n) = A113661(n).

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 + ...

0[ n_] := If[ n < 1, 0, -DivisorSum[ n, (-1)^(n/#) JacobiSymbol[ -3, #] &]]; (* Michael Somos, Feb 19 2015 *)

(PARI) {a(n) = if( n<1, 0, -sumdiv(n, d, (-1)^(n/d) * kronecker(-3, d)))};

(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, (2+(-1)^d) * kronecker(-3, d)))};

(PARI) {a(n) = local(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))};

(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])};

(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==2, (3*(-1)^e - 1) / 2, if(p==3, 1, if(p%6==1, e+1, !(e%2)))))))};

Cf. A113661(n)=-(-1)^n*a(n). A122859(n)=-6*a(n) if n>0.

Cf. A113661, A122859.

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a(3n) = a(4n) = a(n). a(6n+5) = 0.

(PARI) {a(n)= if(n<1, 0, -sumdiv(n, d, (-1)^(n/d)* kronecker(-3, d)))}

(PARI) {a(n)= if(n<1, 0, sumdiv(n, d, (2+(-1)^d)* kronecker(-3, d)))}

(PARI) {a(n)= if(n<1, 0, -sumdiv(n, d, (-1)^(n/d)* kronecker(-3, d)))}

(PARI) {a(n)= if(n<1, 0, sumdiv(n, d, (2+(-1)^d)* kronecker(-3, d)))}

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_Michael Somos, _, Sep 15 2006

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

M. Somos, <a href="http://cis.csuohio.edu/~somos/multiq.pdf">Introduction to Ramanujan theta functions</a>

Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

sign,mult,new

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 84, Eq. (32.64).

G.f.: Sum_{k>0} x^(3*k-2)/(1+x^(3*k-2)) -x^(3*k-1)/(1+x^(3*k-1)).

(PARI) {a(n)= if(n<1, 0, -sumdiv(n, d, (-1)^(n/d)* kronecker(-3, d)))}

(PARI) {a(n)= if(n<1, 0, sumdiv(n, d, (2+(-1)^d)* kronecker(-3, d)))}

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