A113446 Expansion of (phi(q)^2 - phi(q^3)^2) / 4 in powers of q where phi() is a Ramanujan theta function.

1, 1, -1, 1, 2, -1, 0, 1, 1, 2, 0, -1, 2, 0, -2, 1, 2, 1, 0, 2, 0, 0, 0, -1, 3, 2, -1, 0, 2, -2, 0, 1, 0, 2, 0, 1, 2, 0, -2, 2, 2, 0, 0, 0, 2, 0, 0, -1, 1, 3, -2, 2, 2, -1, 0, 0, 0, 2, 0, -2, 2, 0, 0, 1, 4, 0, 0, 2, 0, 0, 0, 1, 2, 2, -3, 0, 0, -2, 0, 2, 1, 2, 0, 0, 4, 0, -2, 0, 2, 2, 0, 0, 0, 0, 0, -1, 2, 1, 0, 3, 2, -2, 0, 2, 0

Offset

1,5

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Michael Somos, Introduction to Ramanujan theta functions, 2019.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Formula

Expansion of (eta(q^2)^3 * eta(q^6) * eta(q^12)^2) / (eta(q) * eta(q^3) * eta(q^4)^2) in powers of q.

Euler transform of period 12 sequence [1, -2, 2, 0, 1, -2, 1, 0, 2, -2, 1, -2, ...].

Moebius transform is period 12 sequence [1, 0, -2, 0, 1, 0, -1, 0, 2, 0, -1, 0, ...].

a(n) is multiplicative and a(2^e) = 1, a(3^e) = (-1)^e, a(p^e) = e+1 if p == 1 (mod 4), a(p^e) = (1 + (-1)^e)/2 if p == 3 (mod 4).

G.f.: ((Sum_{k} x^(k^2))^2 - (Sum_{k} x^(3*k^2))^2) / 4.

G.f.: Sum_{k>0} x^(3*k-1) / (1 + x^(6*k-2)) + x^(3*k-2)/(1 + x^(6*k-4)).

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

G.f.: x * Product_{k>0} (1 - x^k)^2 * (1 + x^k)^3 * (1 + x^(3*k)) * (1 + x^(4*k) + x^(8*k))^2.

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A138949.

a(n) = (-1)^e * A035154(n) where 3^e is the highest power of 3 dividing n.

a(4*n + 1) = A008441(n).

Expansion of q * f(-q, -q^5) * f(q, q^5)^2 / phi(-q^3) in powers of q where phi(), f(,) are Ramanujan theta functions. - Michael Somos, Jan 31 2015

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

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/6 (A019673). - Amiram Eldar, Nov 24 2023

Example

G.f. = q + q^2 - q^3 + q^4 + 2*q^5 - q^6 + q^8 + q^9 + 2*q^10 - q^12 + ...

Mathematica

a[ n_] := If[ n < 1, 0, (-1)^IntegerExponent[ n, 3] Sum[ KroneckerSymbol[ -36, d], { d, Divisors[ n]}]]; (* Michael Somos, Jan 31 2015 *)
a[ n_] := SeriesCoefficient[ (1/4) EllipticTheta[ 2, 0, q^(3/2)]^3 / EllipticTheta[ 2, 0, q^(1/2)] (EllipticTheta[ 3, 0, q] / EllipticTheta[ 3, 0, q^3]), {q, 0, n}]; (* Michael Somos, Jan 31 2015 *)

Prog

(PARI) {a(n) = if( n<1, 0, (-1)^valuation(n, 3) * sumdiv(n, d, kronecker(-36, d)))};
(PARI) {a(n) = if( n<1, 0, direuler(p=2, n, if( p==3, 1 / (1 + X), 1 / (1 - X) / (1 - kronecker(-36, p) * X)))[n])};
(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^6 + A) * eta(x^12 + A)^2 / (eta(x + A) * eta(x^3 + A) * eta(x^4 + A)^2), n))};
(Magma) A := Basis( ModularForms( Gamma1(12), 1), 106); A[2] + A[3] - A[4] + A[5]; /* Michael Somos, Jan 31 2015 */

Crossrefs

Cf. A008441, A019673, A035154, A138949.

Cf. A000122, A000700, A010054, A121373.

Sequence in context: A066295 A132004 A035154 * A121450 A143110 A319662

Adjacent sequences: A113443 A113444 A113445 * A113447 A113448 A113449

Keyword

sign,easy,mult

Author

Michael Somos, Nov 02 2005

Status

approved