A113660 Expansion of phi(x)^3 / phi(x^3) where phi() is a Ramanujan theta function.

1, 6, 12, 6, -6, 0, 12, 12, 12, 6, 0, 0, -6, 12, 24, 0, -6, 0, 12, 12, 0, 12, 0, 0, 12, 6, 24, 6, -12, 0, 0, 12, 12, 0, 0, 0, -6, 12, 24, 12, 0, 0, 24, 12, 0, 0, 0, 0, -6, 18, 12, 0, -12, 0, 12, 0, 24, 12, 0, 0, 0, 12, 24, 12, -6, 0, 0, 12, 0, 0, 0, 0, 12, 12

Offset

0,2

Comments

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).

References

Bruce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, 1991, see p. 227, Entry 4(iv).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Michael Somos, Introduction to Ramanujan theta functions, 2019.

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

Formula

Expansion of a(q) + 2*q(q^2) - 2*a(q^4) = b(-q)^2 / b(q^2) = (b(q) - 2*b(q^4))^2 / b(q^2) = (c(q) + 2*c(q^4))^2 / (3 * c(q^2)) in powers of q where a(), b(), c() are cubic AGM functions. - Michael Somos, May 20 2015

Expansion of (eta(q^2)^15 * eta(q^3)^2 * eta(q^12)^2) / (eta(q)^6 * eta(q^4)^6 * eta(q^6)^5) in powers of q.

a(n) = 6*b(n) where b(n) is multiplicative with a(0) = 1, b(2^e) = (1 - 3(-1)^e) / 2 if e>0, b(3^e) = 1, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6).

Euler transform of period 12 sequence [ 6, -9, 4, -3, 6, -6, 6, -3, 4, -9, 6, -2, ...].

Moebius transform is period 12 sequence [ 6, 6, 0, -18, -6, 0, 6, 18, 0, -6, -6, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 108^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A113973. - Michael Somos, May 20 2015

G.f.: 1 + 6 * ( Sum_{k>0} x^k / (1 + x^k + x^(2*k)) + 2*x^(4*k - 2) / (1 + x^(4*k - 2) + x^(8*k - 4)) ).

a(n) = 6 * A113661(n), if n>0.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi*sqrt(3) = 5.441398... (A304656). - Amiram Eldar, Dec 25 2023

Example

G.f. = 1 + 6*q + 12*q^2 + 6*q^3 - 6*q^4 + 12*q^6 + 12*q^7 + 12*q^8 + 6*q^9 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^3 / EllipticTheta[ 3, 0, q^3], {q, 0, n}]; (* Michael Somos, May 20 2015 *)

Prog

(PARI) {a(n) = my(x); if( n<1, n==0, x = valuation(n, 2); if( n%2, 2, (1 - 3*(-1)^x))*3 * sumdiv(n/2^x, d, kronecker(-3, d)))};
(PARI) {a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); 6*prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, (1 - 3*(-1)^e) / 2, p==3, 1, p%6==1, e+1, !(e%2))))};
(PARI) {a(n) = if( n<1, n==0, 6 * direuler(p=2, n, if( p==2, 2 - (1 - 2*X) / (1 - X^2), 1 / ((1 - X) * (1 - kronecker(-3, p) * X))))[n])};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^15 * eta(x^3 + A)^2 * eta(x^12 + A)^2 / (eta(x + A)^6 * eta(x^4 + A)^6 * eta(x^6 + A)^5), n))};
(Magma) A := Basis( ModularForms( Gamma1(12), 1), 74); A[1] + 6*A[2] + 12*A[3] + 6*A[4] - 6*A[5]; /* Michael Somos, May 20 2015 */

Crossrefs

Cf. A113661, A113973, A304656.

Cf. A000122, A000700, A004016, A005882, A005928, A010054, A121373.

Sequence in context: A356119 A263538 A076590 * A122859 A315773 A315774

Adjacent sequences: A113657 A113658 A113659 * A113661 A113662 A113663

Keyword

sign,easy

Author

Michael Somos, Nov 03 2005

Extensions

Corrected by Charles R Greathouse IV, Sep 02 2009

Status

approved