%I #15 Feb 16 2025 08:33:19
%S 1,1,2,4,6,9,16,21,32,48,66,92,134,177,243,334,441,585,788,1018,1334,
%T 1746,2240,2877,3715,4698,5972,7582,9517,11940,15005,18639,23190,
%U 28812,35544,43808,53996,66084,80887,98868,120278,146157,177497,214562,259227
%N Expansion of x * f(-x^7) * f(-x^21) / (f(-x) * f(-x^3)) where f() is a Ramanujan theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A226007/b226007.txt">Table of n, a(n) for n = 1..2500</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="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of eta(q^7) * eta(q^21) / (eta(q) * eta(q^3)) in powers of q.
%F Euler transform of period 21 sequence [1, 1, 2, 1, 1, 2, 0, 1, 2, 1, 1, 2, 1, 0, 2, 1, 1, 2, 1, 1, 0, ...].
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u*v * (1 + 7 * u*v) - (u+v) * (u^2 - 3 * u*v + v^2).
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (21 t)) = 1/7 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A226006.
%F G.f.: x * Product_{k>0} (1 - x^(7*k)) * (1 - x^(21*k)) / ( (1 - x^k) * (1 - x^(3*k))).
%F Convolution inverse of A226006.
%F a(n) ~ exp(4*Pi*sqrt(n/21)) / (sqrt(2) * 3^(1/4) * 7^(5/4) * n^(3/4)). - _Vaclav Kotesovec_, Oct 14 2015
%e G.f. = q + q^2 + 2*q^3 + 4*q^4 + 6*q^5 + 9*q^6 + 16*q^7 + 21*q^8 + 32*q^9 + ...
%t a[ n_] := SeriesCoefficient[ q QPochhammer[ q^7] QPochhammer[ q^21] / (QPochhammer[ q] QPochhammer[ q^3]), {q, 0, n}]; (* _Michael Somos_, Apr 12 2015 *)
%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^7 + A) * eta(x^21 + A) / (eta(x + A) * eta(x^3 + A)), n))};
%Y Cf. A226006.
%K nonn,changed
%O 1,3
%A _Michael Somos_, May 22 2013