login
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.
8

%I #16 Apr 24 2023 15:58:40

%S 1,1,4,10,20,39,76,140,244,415,696,1140,1820,2861,4448,6816,10292,

%T 15372,22756,33356,48408,69683,99600,141312,199036,278557,387608,

%U 536230,737632,1009464,1374888,1863764,2514868,3378948,4521672,6027000,8002676

%N 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.

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

%H G. C. Greubel, <a href="/A164617/b164617.txt">Table of n, a(n) for n = 0..1000</a>

%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F 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.

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

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

%F a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(9/4) * n^(3/4)). - _Vaclav Kotesovec_, Oct 13 2015

%F A128641(n) = (-1)^n*a(n). - _Michael Somos_, Apr 24 2023

%e 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 + ...

%t 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 *)

%o (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))};

%Y Cf. A113973, A128641, A132974, A164616.

%K nonn

%O 0,3

%A _Michael Somos_, Aug 17 2009