A227587 Expansion of (phi(-q^3)^2 / (phi(-q) * phi(-q^9)))^2 in powers of q where phi() is a Ramanujan theta function.

1, 4, 12, 24, 44, 72, 120, 192, 300, 456, 680, 1008, 1464, 2104, 2976, 4176, 5804, 7992, 10920, 14800, 19944, 26688, 35504, 46944, 61752, 80828, 105288, 136536, 176288, 226728, 290448, 370720, 471468, 597600, 755032, 950976, 1194216, 1495352, 1867344, 2325648

Offset

0,2

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..2500

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.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of 1 + 4 * q * chi(-q^3)^2 / (chi(-q) * chi(-q^9))^3 in powers of q where chi() is a Ramanujan theta function.

Expansion of 1 + 4 * b(q^2) * c(q^6) / (b(q) * c(q^3)) in powers of q where b(), c() are cubic AGM theta functions.

Expansion of (eta(q^2) * eta(q^3)^4 * eta(q^18) / (eta(q) * eta(q^6) * eta(q^9))^2)^2 in powers of q.

Euler transform of period 18 sequence [4, 2, -4, 2, 4, -2, 4, 2, 0, 2, 4, -2, 4, 2, -4, 2, 4, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. for A215412.

a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (2^(3/4) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Jul 11 2016

Example

G.f. = 1 + 4*q + 12*q^2 + 24*q^3 + 44*q^4 + 72*q^5 + 120*q^6 + 192*q^7 + 300*q^8 + ...

Mathematica

nmax = 50; CoefficientList[Series[Product[((1-x^(2*k)) * (1-x^(3*k))^4 * (1-x^(18*k)) / ((1-x^k) * (1-x^(6*k)) * (1-x^(9*k)))^2)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 11 2016 *)
eta[q_] := q^(1/24)*QPochhammer[q]; CoefficientList[Series[(eta[q^2]* eta[q^3]^4*eta[q^18]/(eta[q]*eta[q^6]*eta[q^9])^2)^2, {q, 0, 50}], q] (* G. C. Greubel, Aug 09 2018 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A)^4 * eta(x^18 + A) / (eta(x + A) * eta(x^6 + A) * eta(x^9 + A))^2)^2, n))};

Crossrefs

Cf. A215412.

Sequence in context: A301211 A008195 A001209 * A128624 A372435 A321879

Adjacent sequences: A227584 A227585 A227586 * A227588 A227589 A227590

Keyword

nonn

Author

Michael Somos, Jul 16 2013

Status

approved