A123629 Expansion of b(q^2) * c(q^6) / (b(q) * c(q^3)) in powers of q where b(), c() are cubic AGM theta functions.

1, 3, 6, 11, 18, 30, 48, 75, 114, 170, 252, 366, 526, 744, 1044, 1451, 1998, 2730, 3700, 4986, 6672, 8876, 11736, 15438, 20207, 26322, 34134, 44072, 56682, 72612, 92680, 117867, 149400, 188758, 237744, 298554, 373838, 466836, 581412, 722266, 895014

Offset

1,2

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

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

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015

Formula

Expansion of (eta(q^3) / eta(q^6))^2 * (eta(q^2) * eta(q^18) / (eta(q) * eta(q^9)))^3 in powers of q.

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

Euler transform of period 18 sequence [ 3, 0, 1, 0, 3, 0, 3, 0, 4, 0, 3, 0, 3, 0, 1, 0, 3, 0, ...].

Convolution inverse is A123676. - Michael Somos, Feb 19 2015

Expansion of q * chi(-q^3)^2 / (chi(-q) * chi(-q^9))^3 in powers of q where chi() is a Ramanujan theta function. - Michael Somos, Feb 19 2015

G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = (1/4) * g(t) where q = exp(2 Pi i t) and g() is the g.f. for A123676. - Michael Somos, Feb 19 2015

a(3*n) = 6 * A128638(n). a(3*n + 2) = 3 * A233698(n). - Michael Somos, Feb 19 2015

a(n) ~ exp(2*sqrt(2*n)*Pi/3) / (2^(11/4)*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Oct 10 2015

Example

G.f. = q + 3*q^2 + 6*q^3 + 11*q^4 + 18*q^5 + 30*q^6 + 48*q^7 + 75*q^8 + ...

Mathematica

nmax=60; CoefficientList[Series[Product[(1+x^k)^3 * (1+x^(9*k))^3 / (1+x^(3*k))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 10 2015 *)
A123629[n_] := SeriesCoefficient[q*(QPochhammer[q^3]/QPochhammer[q^6])^2*(QPochhammer[q^2]*QPochhammer[q^18]/(QPochhammer[q]*QPochhammer[q^9] ))^3, {q, 0, n}]; Table[A123629[n], {n, 0, 50}] (* G. C. Greubel, Oct 09 2017 *)

Prog

(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^3 + A) / eta(x^6 + A))^2 * (eta(x^2 + A) * eta(x^18 + A) / (eta(x + A) * eta(x^9 + A)))^3, n))};

Crossrefs

Cf. A123676, A128638, A233698.

Sequence in context: A147079 A281572 A152074 * A212484 A279100 A347415

Adjacent sequences: A123626 A123627 A123628 * A123630 A123631 A123632

Keyword

nonn

Author

Michael Somos, Oct 03 2006

Extensions

Typo in xrefs corrected by Vaclav Kotesovec, Oct 10 2015

Status

approved