A128758 Expansion of q^(-1/3) * (eta(q^3) / eta(q))^4 in powers of q.

1, 4, 14, 36, 89, 196, 416, 828, 1600, 2972, 5390, 9504, 16436, 27828, 46364, 75960, 122772, 195728, 308430, 480456, 740921, 1131364, 1712348, 2569500, 3825641, 5652872, 8294612, 12089016, 17508609, 25204428, 36076540, 51355368, 72725909

Offset: 0

Author

Michael Somos, Mar 24 2007

Keywords

• nonn

Comments

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

Examples

			G.f. = 1 + 4*x + 14*x^2 + 36*x^3 + 89*x^4 + 196*x^5 + 416*x^6 + 828*x^7 + ...
G.f. = q + 4*q^4 + 14*q^7 + 36*q^10 + 89*q^13 + 196*q^16 + 416*q^19 + ...
		

References

• O. Kolberg, The coefficients of j(tau) modulo powers of 3, Acta Univ. Bergen., Series Math., Arbok for Universitetet I Bergen, Mat.-Naturv. Serie, 1962 No. 16, pp. 1-7. See v, page 1.

Links

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

Crossrefs

Cf. A112146.

Programs

Mathematica

nmax = 40; CoefficientList[Series[Product[((1-x^(3*k)) / (1-x^k))^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 07 2015 *)
eta[q_]:= q^(1/24)*QPochhammer[q]; c:= q^(1/3)*(eta[q]/eta[q^3])^4; a:= CoefficientList[Series[1/c, {q,0,60}], q]; Table[a[[n]], {n,1,50}] (* G. C. Greubel, Jul 04 2018 *)

PARI

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^3 + A) / eta(x + A))^4, n))};

Formula

Expansion of q^(-1/3) * (1/3) * c(q) / b(q) in powers of q where b(), c() are cubic AGM theta functions.

Euler transform of period 3 sequence [ 4, 4, 0, ...].

Given g.f. A(x), then B(q) = q*A(q^3) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (u+v)^3 - u*v * (1+3*u) * (1+3*v).

Given g.f. A(x), then B(q)= q*A(q^3) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u^2 + w^2 + u*w - v - 9*v^2 * (u+w).

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

9*a(n) = A112146(3*n + 1).

a(n) ~ exp(4*Pi*sqrt(n)/3) / (9*sqrt(6)*n^(3/4)). - Vaclav Kotesovec, Sep 07 2015