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A258040 Expansion of f(x) / f(-x) in powers of x where f() is the g.f. for A007325.

%I A258040 #10 Feb 16 2025 08:33:25
%S A258040 1,-2,2,0,-2,2,0,0,-2,2,2,-8,8,0,-8,8,-2,0,-6,8,6,-24,24,0,-24,22,-4,
%T A258040 0,-16,20,16,-64,62,0,-60,56,-10,0,-40,48,38,-148,144,0,-136,126,-24,
%U A258040 0,-88,106,82,-320,308,0,-288,264,-48,0,-180,216,168,-652,624
%N A258040 Expansion of f(x) / f(-x) in powers of x where f() is the g.f. for A007325.
%H A258040 G. C. Greubel, <a href="/A258040/b258040.txt">Table of n, a(n) for n = 0..1000</a>
%H A258040 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A258040 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A258040 Expansion of f(-x, -x^4) * f(-x^2, +x^3) / (f(+x, -x^4) * f(-x^2, -x^3)) = f(-x, -x^9) * f(+x^3, +x^7) / (f(+x, +x^9) * f(-x^3, -x^7)) in powers of x where f(,) is the Ramanujan general theta function.
%F A258040 Euler transform of period 20 sequence [ -2, 1, 2, 0, 0, -1, 2, 0, -2, 0, -2, 0, 2, -1, 0, 0, 2, 1, -2, 0, ...].
%F A258040 a(10*n + 3) = a(10*n + 7) = 0.
%e A258040 G.f. = 1 - 2*x + 2*x^2 - 2*x^4 + 2*x^5 - 2*x^8 + 2*x^9 + 2*x^10 - 8*x^11 + ...
%t A258040 a[ n_] := SeriesCoefficient[ Product[(1 - x^k)^{ 2, -1, -2, 0, 0, 1, -2, 0, 2, 0, 2, 0, -2, 1, 0, 0, -2, -1, 2, 0}[[ Mod[k, 20, 1]]], {k, 1, n}], {x, 0, n}];
%o A258040 (PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^ [ 0, 2, -1, -2, 0, 0, 1, -2, 0, 2, 0, 2, 0, -2, 1, 0, 0, -2, -1, 2][k%20 + 1]) ,n))};
%Y A258040 Cf. A007325.
%K A258040 sign
%O A258040 0,2
%A A258040 _Michael Somos_, May 16 2015