A106401 Expansion of (eta(q) * eta(q^9))^3 / eta(q^3)^2 in powers of q.

1, -3, 0, 7, -6, 0, 8, -15, 0, 18, -12, 0, 14, -24, 0, 31, -18, 0, 20, -42, 0, 36, -24, 0, 31, -42, 0, 56, -30, 0, 32, -63, 0, 54, -48, 0, 38, -60, 0, 90, -42, 0, 44, -84, 0, 72, -48, 0, 57, -93, 0, 98, -54, 0, 72, -120, 0, 90, -60, 0, 62, -96, 0, 127, -84

Offset

1,2

Comments

Number 21 of the 74 eta-quotients listed in Table I of Martin (1996).

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

Links

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

Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.

Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers, 2016.

Formula

G.f.: x * Product_{k>0} (1 - x^k)^3 * (1 - x^(9*k))^3 / (1 - x^(3*k))^2.

Expansion of b(q) * c(q^3) / 3 in powers of q where b(), c() are cubic AGM theta functions. - Michael Somos, Oct 17 2006

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = -v^3 + 6 * u*v*w + 4 * u*w^2 + u^2*w.

Euler transform of period 9 sequence [ -3, -3, -1, -3, -3, -1, -3, -3, -4, ...].

a(n) is multiplicative with a(3^e) = 0 if e>0, a(p^e) = (p^(e+1) - 1) / (p - 1) if e even or p == 1 (mod 3), a(p^e) = -(p^(e+1) - 1) / (p - 1) otherwise. - Michael, Somos Oct 19 2005

G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = 9 (t/i)^2 f(t) where q = exp(2 Pi i t).

a(3*n) = 0. a(3*n + 1) = A144614(n). a(3*n + 2) = -3 * A033686(n).

Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = 4*Pi^2/81 = 0.487387... . - Amiram Eldar, Jan 23 2024

Example

G.f. = q - 3*q^2 + 7*q^4 - 6*q^5 + 8*q^7 - 15*q^8 + 18*q^10 - 12*q^11 +...

Mathematica

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^9])^3 / QPochhammer[ q^3]^2, {q, 0, n}]; (* Michael Somos, May 18 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^9 + A))^3 / eta(x^3 + A)^2, n))};
(PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, 0, (-1)^((p%3>1) * e) * (p^(e+1) - 1) / (p - 1))))};
(Magma) A := Basis( ModularForms( Gamma0(9), 2), 66); A[2] - 3*A[3]; /* Michael Somos, May 18 2015 */

Crossrefs

Cf. A033686, A144614.

Cf. A004016, A005882, A005928.

Sequence in context: A104687 A298097 A155831 * A332329 A011200 A201573

Adjacent sequences: A106398 A106399 A106400 * A106402 A106403 A106404

Keyword

sign,mult

Author

Michael Somos, May 02 2005

Status

approved