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A208605 Expansion of q * psi(q^8) / phi(q) in powers of q where phi(), psi() are Ramanujan theta functions.

%I A208605 #19 Feb 16 2025 08:33:16
%S A208605 1,-2,4,-8,14,-24,40,-64,101,-156,236,-352,518,-752,1080,-1536,2162,
%T A208605 -3018,4180,-5744,7840,-10632,14328,-19200,25591,-33932,44776,-58816,
%U A208605 76918,-100176,129952,-167936,216240,-277476,354864,-452392,574958,-728568,920600
%N A208605 Expansion of q * psi(q^8) / phi(q) in powers of q where phi(), psi() are Ramanujan theta functions.
%C A208605 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A208605 G. C. Greubel, <a href="/A208605/b208605.txt">Table of n, a(n) for n = 1..1000</a>
%H A208605 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A208605 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A208605 Expansion of eta(q)^2 * eta(q^4)^2 * eta(q^16)^2 / (eta(q^2)^5 * eta(q^8)) in powers of q.
%F A208605 Euler transform of period 16 sequence [ -2, 3, -2, 1, -2, 3, -2, 2, -2, 3, -2, 1, -2, 3, -2, 0, ...].
%F A208605 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 g.f. for A208603.
%F A208605 a(n) = -(-1)^n * A123655(n). a(2*n) = -2 * A107035(n). a(2*n + 1) = A093160(n). Convolution inverse of A208603.
%e A208605 q - 2*q^2 + 4*q^3 - 8*q^4 + 14*q^5 - 24*q^6 + 40*q^7 - 64*q^8 + 101*q^9 + ...
%t A208605 eta[q_] := q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[eta[q]^2* eta[q^4]^2*eta[q^16]^2/(eta[q^2]^5*eta[q^8]), {q, 0, n}]; Table[a[n], {n,1,50}] (* _G. C. Greubel_, Jan 23 2018 *)
%o A208605 (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^16 + A)^2 / (eta(x^2 + A)^5 * eta(x^8 + A)), n))}
%Y A208605 Cf. A093160, A107035, A123655, A208603.
%K A208605 sign
%O A208605 1,2
%A A208605 _Michael Somos_, Feb 29 2012