0,2

L.-C. Shen, On the Modular Equations of Degree 3. Proc. Amer. Math. Soc. 122 (1994), no. 4, 1101-1114. See p. 1108, Eq. (3.20).

Expansion of (3 * phi(q^3)^2 - phi(q)^2) / 2 in powers of q where phi() is a Ramanujan theta function.

Expansion of phi(-q) * phi(-q^2) * chi(q^3) / chi(-q^3) in powers of q where phi(), chi() are Ramanujan theta functions.

Euler transform of period 12 sequence [ -2, -3, 0, -2, -2, -4, -2, -2, 0, -3, -2, -2, ...].

Moebius transform is period 12 sequence [ -2, 0, 8, 0, -2, 0, 2, 0, -8, 0, 2, 0, ...].

a(n) = -2 * b(n) where b(n) is multiplicative and b(2^e) = 1, b(3^e) = -1 + 2 * (-1)^e, b(p^e) = e+1 if p == 1, 5 (mod 12), b(p^e) = (1 + (-1)^e) / 2 if p == 7, 11 (mod 12).

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 12 (t/i) g(t) where q = exp(2 pi i t) and g() is g.f. for A113446.

a(12*n + 7) = a(12*n + 11) = 0. a(2*n) = a(n).

G.f.: Product_{k>0} (1 - x^(2*k))^2 * (1 - x^k + x^(2*k))^2 / ((1 + x^(2*k))^2 * (1 - x^(2*k) + x^(4*k))).

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

1 - 2*q - 2*q^2 + 6*q^3 - 2*q^4 - 4*q^5 + 6*q^6 - 2*q^8 - 2*q^9 + ...

(PARI) {a(n) = if( n<1, n==0, 2 * sumdiv(n, d, kronecker(-4, n/d) * [2, -1, -1][d%3 + 1]))}

(PARI) {a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); -2 * prod( k=1, matsize(A)[1], if( p = A[k, 1], e = A[k, 2]; if( p==2, 1, if( p==3, -1 + 2 * (-1)^e, if(p%12 < 6, e+1, (1 + (-1)^e) / 2)))))) }

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^2 + A) * eta(x^6 + A)^3 / (eta(x^3 + A)^2 * eta(x^4 + A) * eta(x^12 + A)), n))}

-2 * A138950(n) = a(n) unless n=0. -2 * A116604(n) = a(2*n + 1). -2 * A122865(n) = a(3*n + 1). -2 * A122856(n) = a(3*n + 2). -2 * A008441(n) = a(4*n + 1).

Sequence in context: A124859 A021446 A062401 this_sequence A138951 A163370 A071796

Adjacent sequences: A138946 A138947 A138948 this_sequence A138950 A138951 A138952

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Michael Somos, Apr 03 2008