A164617 Expansion of (phi^3(q^3) / phi(q)) * (psi(-q^3) / psi^3(-q)) in powers of q where phi(), psi() are Ramanujan theta functions.

1, 1, 4, 10, 20, 39, 76, 140, 244, 415, 696, 1140, 1820, 2861, 4448, 6816, 10292, 15372, 22756, 33356, 48408, 69683, 99600, 141312, 199036, 278557, 387608, 536230, 737632, 1009464, 1374888, 1863764, 2514868, 3378948, 4521672, 6027000, 8002676

Offset

0,3

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..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

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of eta(q^6)^14 / (eta(q) * eta(q^2)^2 * eta(q^3)^5 * eta(q^4) * eta(q^12)^5) in powers of q.

Euler transform of period 12 sequence [ 1, 3, 6, 4, 1, -6, 1, 4, 6, 3, 1, 0, ...].

Convolution of A113973 and A132974. a(n) = A164616(3*n).

a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(9/4) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015

A128641(n) = (-1)^n*a(n). - Michael Somos, Apr 24 2023

Example

G.f. = 1 + q + 4*q^2 + 10*q^3 + 20*q^4 + 39*q^5 + 76*q^6 + 140*q^7 + 244*q^8 + ...

Mathematica

nmax=60; CoefficientList[Series[Product[(1-x^(6*k))^14 / ((1-x^k) * (1-x^(2*k))^2 * (1-x^(3*k))^5 * (1-x^(4*k)) * (1-x^(12*k))^5), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 13 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^14 / (eta(x + A) * eta(x^2 + A)^2 * eta(x^3 + A)^5 * eta(x^4 + A) * eta(x^12 + A)^5), n))};

Crossrefs

Cf. A113973, A128641, A132974, A164616.

Sequence in context: A258346 A128640 A128641 * A366646 A038421 A049032

Adjacent sequences: A164614 A164615 A164616 * A164618 A164619 A164620

Keyword

nonn

Author

Michael Somos, Aug 17 2009

Status

approved