Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.
%I #13 Mar 12 2021 22:24:48
%S 1,2,6,14,30,60,114,208,366,626,1044,1704,2730,4300,6672,10212,15438,
%T 23076,34134,50008,72612,104560,149400,211920,298554,417902,581412,
%U 804254,1106448,1514316,2062332,2795488,3772302,5068632,6782508,9040224,12004014,15881692
%N Expansion of f(x^2, x^4) * f(x^3, x^3) / f(-x, -x^2)^2 in power of x where f(, ) is Ramanujan's general 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="/A260058/b260058.txt">Table of n, a(n) for n = 0..1000</a>
%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^4) * eta(q^6)^7 / (eta(q)^2 * eta(q^2) * eta(q^3)^2 * eta(q^12)^3) in powers of q.
%F Euler transform of period 12 sequence [ 2, 3, 4, 2, 2, -2, 2, 2, 4, 3, 2, 0, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (1/6) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A187145.
%F a(n) = A261154(3*n).
%F a(n) ~ exp(2*Pi*sqrt(n/3)) / (4*3^(5/4)*n^(3/4)). - _Vaclav Kotesovec_, Mar 17 2018
%e G.f. = 1 + 2*x + 6*x^2 + 14*x^3 + 30*x^4 + 60*x^5 + 114*x^6 + 208*x^7 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x^2, x^6] QPochhammer[ -x^4, x^6] QPochhammer[ x^6] EllipticTheta[ 3, 0, x^3] / QPochhammer[ x]^2, {x, 0, n}];
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A) * eta(x^6 + A)^7 / (eta(x + A)^2 * eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^12 + A)^3), n))};
%Y Cf. A261154.
%K nonn
%O 0,2
%A _Michael Somos_, Nov 08 2015