A121589 Series expansion of (eta(q^9) / eta(q))^3 in powers of q.

1, 3, 9, 22, 51, 108, 221, 429, 810, 1476, 2631, 4572, 7802, 13056, 21519, 34918, 55935, 88452, 138332, 213990, 327852, 497592, 748833, 1117692, 1655719, 2434938, 3556791, 5161808, 7445631, 10677096, 15226658, 21599469, 30485268, 42817788

Offset: 1

Author

Michael Somos, Aug 09 2006

Keywords

• nonn

Comments

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

Examples

			G.f. = q + 3*q^2 + 9*q^3 + 22*q^4 + 51*q^5 + 108*q^6 + 221*q^7 + 429*q^8 + ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000
• Kevin Acres, David Broadhurst, Eta quotients and Rademacher sums, arXiv:1810.07478 [math.NT], 2018. See Table 1 p. 10.

Crossrefs

Cf. A104502, A131986.

Programs

Maple

N:= 100: # to get a(1)..a(N)
S:= series(q*Product(1-q^(9*k),k=1..N/9)/Product((1-q^k)^3, k=1..N),q,N+1):
seq(coeff(S,q,n),n=1..N); # Robert Israel, Nov 02 2017

Mathematica

nmax = 40; CoefficientList[Series[Product[((1-x^(9*k))/(1-x^k))^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 07 2015 *)
QP = QPochhammer; s = (QP[q^9]/QP[q])^3 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, adapted from PARI *)
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^9] / QPochhammer[ q])^3, {q, 0, n}]; (* Michael Somos, Nov 02 2017 *)

PARI

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^9 + A) / eta(x + A))^3, n))};

Formula

Euler transform of period 9 sequence [3, 3, 3, 3, 3, 3, 3, 3, 0, ...].

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

Expansion of c(q^3) / (3 * b(q)) = (c(q) / (3 * b(q^3))^3 in powers of q where b(), c() are cubic AGM functions.

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u - v^2) * (u^2 - v) - 2 * u * v * ( 3 * (u + v) + 13 * u * v ).

G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u^3 - v * (1 + 9 * v + 27 * v^2) * (1 + 9 * u + 27 * u^2).

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 - (1 + 3 * (u1 + u2)) * (u3 + u6 + 9 * u3 * u6).

G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = (1/27) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A131986.

a(n) ~ exp(4*Pi*sqrt(n)/3) / (27 * sqrt(6) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015

a(1) = 1, a(n) = (3/(n-1))*Sum_{k=1..n-1} A116607(k)*a(n-k) for n > 1. - Seiichi Manyama, Apr 01 2017

Convolution inverse of A131986. Convolution cube of A104502. - Michael Somos, Nov 02 2017

Extensions

Second formula corrected by Vaclav Kotesovec, Sep 07 2015