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

0,2

Comments

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

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

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

Example

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 + ...

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 *)

Prog

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

Crossrefs

Cf. A112146.

Sequence in context: A187091 A034528 A332834 * A341384 A258343 A317148

Adjacent sequences: A128755 A128756 A128757 * A128759 A128760 A128761

Keyword

nonn

Author

Michael Somos, Mar 24 2007

Status

approved