|
%I #13 Mar 12 2021 22:24:48
%S 1,-4,12,-32,80,-184,400,-832,1664,-3220,6056,-11104,19904,-34968,
%T 60320,-102336,171008,-281800,458428,-736928,1171552,-1843328,2872368,
%U -4435392,6790656,-10313180,15544136,-23259968,34568576,-51042392,74901984,-109268224,158507008
%N Expansion of (phi(q^4) / phi(q))^2 in powers of q where phi() 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="/A260186/b260186.txt">Table of n, a(n) for n = 0..2500</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) / eta(q^16))^4 * (eta(q^8) / eta(q^2))^10 in powers of q.
%F Euler transform of period 16 sequence [ -4, 6, -4, 6, -4, 6, -4, -4, -4, 6, -4, 6, -4, 6, -4, 0, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = (1/4) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A216060.
%F Convolution inverse is A216060. Convolution square of A112128.
%F a(n) ~ (-1)^n * exp(Pi*sqrt(2*n)) / (32 * 2^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Nov 15 2017
%e G.f. = 1 - 4*x + 12*x^2 - 32*x^3 + 80*x^4 - 184*x^5 + 400*x^6 - 832*x^7 + ...
%t a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q^4] / EllipticTheta[ 3, 0, q])^2, {q, 0, n}];
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^16 + A))^4 * (eta(x^8 + A) / eta(x^2 + A))^10, n))};
%Y Cf. A112128, A216060.
%K sign
%O 0,2
%A _Michael Somos_, Jul 17 2015
|
