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%I #10 Mar 12 2021 22:24:47
%S 1,3,1,-4,0,9,-1,-20,1,38,1,-64,-2,107,-2,-180,3,292,4,-452,-4,686,-5,
%T -1044,5,1563,6,-2276,-8,3284,-9,-4724,12,6712,13,-9408,-14,13086,-17,
%U -18112,18,24879,21,-33864,-26,45806,-28,-61696,34,82614,39,-109892
%N Expansion of (phi(x) / phi(x^2)) * (f(-x^3, -x^5) / f(-x^1, -x^7)) in powers of x where phi(), f() are Ramanujan theta functions.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A245434/b245434.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 Euler transform of period 8 sequence [3, -5, 1, 2, 1, -5, 3, 0, ...].
%F a(n) = A245436(2*n - 1). a(2*n) = A245433(n).
%e G.f. = 1 + 3*x + x^2 - 4*x^3 + 9*x^5 - x^6 - 20*x^7 + x^8 + 38*x^9 + ...
%e G.f. = 1/q + 3*q + q^3 - 4*q^5 + 9*q^9 - q^11 - 20*q^13 + q^15 + 38*q^17 + ...
%t f[x_, y_]:= QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; a:= CoefficientList[Series[(f[q, q]/f[q^2, q^2])*(f[-q^3, -q^5]/f[-q, -q^7]), {q, 0, 60}], q]]; Table[a[[n]], {n, 1, 50}] (* _G. C. Greubel_, Aug 06 2018 *)
%o (PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[0, -3, 5, -1, -2, -1, 5, -3][k%8 + 1]), n))};
%Y Cf. A245433, A245436.
%K sign
%O 0,2
%A _Michael Somos_, Jul 21 2014
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