cp's OEIS Frontend

A210065 Expansion of phi(q^2) / phi(q) in powers of q where phi() is a Ramanujan theta function.

%I A210065 #19 Feb 16 2025 08:33:17
%S A210065 1,-2,6,-12,22,-40,68,-112,182,-286,440,-668,996,-1464,2128,-3056,
%T A210065 4342,-6116,8538,-11820,16248,-22176,30068,-40528,54308,-72378,95976,
%U A210065 -126648,166352,-217560,283344,-367552,474998,-611624,784812,-1003712,1279562,-1626216
%N A210065 Expansion of phi(q^2) / phi(q) in powers of q where phi() is a Ramanujan theta function.
%C A210065 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A210065 G. C. Greubel, <a href="/A210065/b210065.txt">Table of n, a(n) for n = 0..2500</a>
%H A210065 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A210065 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A210065 Expansion of (eta(q) / eta(q^8))^2 * (eta(q^4) / eta(q^2))^7 in powers of q.
%F A210065 Euler transform of period 8 sequence [-2, 5, -2, -2, -2, 5, -2, 0, ...].
%F A210065 G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 2^(-1/2) * g(t) where q = exp(2 Pi i t) and g() is g.f. for A080015.
%F A210065 a(n) = (-1)^n * A208850(n). Convolution inverse of A080015.
%F A210065 a(n) ~ (-1)^n * exp(sqrt(n)*Pi) / (8*n^(3/4)). - _Vaclav Kotesovec_, Nov 17 2017
%e A210065 G.f. = 1 - 2*q + 6*q^2 - 12*q^3 + 22*q^4 - 40*q^5 + 68*q^6 - 112*q^7 + 182*q^8 + ...
%t A210065 nmax = 40; CoefficientList[Series[Product[((1 - x^k) / (1 - x^(8*k)))^2 * (1 + x^(2*k))^7, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Nov 17 2017 *)
%t A210065 eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[(eta[q]/ eta[q^8])^2*(eta[q^4]/eta[q^2])^7, {q, 0, 50}], q] (* _G. C. Greubel_, Aug 11 2018 *)
%o A210065 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^8 + A))^2 * (eta(x^4 + A) / eta(x^2 + A))^7, n))};
%Y A210065 Cf. A080015, A208850.
%K A210065 sign
%O A210065 0,2
%A A210065 _Michael Somos_, Mar 16 2012