A122859 Expansion of phi(-q)^3 / phi(-q^3) in powers of q 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, 24, -6, -12, 0, 24, -12, 0, -6, 0, 0, -12, 0, 24

Offset

0,2

Comments

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

References

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

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 2*a(q^2) - a(q) = b(q)^2 / b(q^2) in powers of q where a(), b() are cubic AGM theta functions.

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

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

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

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v*(u+v)^2 - 2*u*w*(v+w).

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = (u1-u2-u3+u6) * (u1+2*u2+u3) - (2*u1+u2-2*u3-u6) * (u1+2*u2-u3).

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

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

/ (1 + x^(3*k - 1))).

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

(-1)^n * a(n) = A113660(n). -6 * a(n) = A122860(n) if n>0.

a(2*n) = A227354(n). a(2*n + 1) = -6 * A033762(n). a(3*n + 1) = -6 * A033687(n). a(4*n + 1) = -6 * A112604(n). a(4*n + 3) = -6 * A112605(n). a(6*n + 1) = -6 * A097195(n). a(8*n + 1) = -6 * A112606(n). a(8*n + 3) = -6 * A112608(n). a(8*n + 5) = -12 * A112607(n-1). a(8*n + 7) = -12 * A112609(n). a(12*n + 1) = -6 * A123884(n). a(12*n + 7) = -12 * A121361(n). - Michael Somos, Sep 27 2013

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 0. - Amiram Eldar, Nov 23 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[ 4, 0, q]^3 / EllipticTheta[ 4, 0, q^3], {q, 0, n}] (* Michael Somos, Sep 27 2013 *)

Prog

(PARI) {a(n)= if( n<1, n==0, 6 * sumdiv(n, d, (-1)^(n/d) * kronecker( -3, d)))}
(PARI) {a(n)= if( n<1, n==0, -6 * sumdiv(n, d, (2 + (-1)^d) * kronecker( -3, d)))}
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^6 * eta(x^6 + A) / (eta(x^2 + A)^3 * eta(x^3 + A)^2), n))}
(Sage) A = ModularForms( Gamma1(6), 1, prec=90).basis(); A[0] - 6 *A[1] # Michael Somos, Sep 27 2013

Crossrefs

Cf. A033687, A033762, A097195, A112604, A112605, A112606, A112607, A112608, A112609, A113660, A121361, A122860, A123884, A227354.

Cf. A000122, A000700, A010054, A121373.

Sequence in context: A263538 A076590 A113660 * A315773 A315774 A322214

Adjacent sequences: A122856 A122857 A122858 * A122860 A122861 A122862

Keyword

sign

Author

Michael Somos, Sep 15 2006

Status

approved