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A128758 Expansion of q^(-1/3) * (eta(q^3) / eta(q))^4 in powers of q. 6
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 (list; graph; refs; listen; history; text; internal format)
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: A193522 A187091 A034528 * A258343 A317148 A027166

Adjacent sequences:  A128755 A128756 A128757 * A128759 A128760 A128761

KEYWORD

nonn

AUTHOR

Michael Somos, Mar 24 2007

STATUS

approved

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Last modified January 21 17:55 EST 2019. Contains 319349 sequences. (Running on oeis4.)