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A226559 Expansion of f(-x^1, -x^7) * f(-x^2, -x^6) / (f(-x^3, -x^5) * f(-x^4, -x^4)) in powers of x where f(, ) is Ramanujan's general theta function.

%I A226559 #19 Feb 16 2025 08:33:20
%S A226559 1,-1,-1,2,1,-2,-2,2,4,-4,-5,6,6,-7,-9,10,13,-15,-17,20,21,-25,-28,32,
%T A226559 39,-43,-49,56,60,-69,-78,86,101,-112,-125,142,153,-172,-192,212,241,
%U A226559 -266,-295,328,357,-397,-438,482,540,-592,-652,720,781,-862,-946
%N A226559 Expansion of f(-x^1, -x^7) * f(-x^2, -x^6) / (f(-x^3, -x^5) * f(-x^4, -x^4)) in powers of x where f(, ) is Ramanujan's general theta function.
%C A226559 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A226559 G. C. Greubel, <a href="/A226559/b226559.txt">Table of n, a(n) for n = 0..1000</a>
%H A226559 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A226559 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A226559 Expansion of (f(-x^1, -x^7) / f(-x^3, -x^5)) * (psi(-x^2) / phi(-x^4)) in powers of x where psi(), phi(), f() are Ramanujan theta functions.
%F A226559 Euler transform of period 8 sequence [-1, -1, 1, 2, 1, -1, -1, 0, ...].
%F A226559 Given g.f. A(x) then B(q) = q^3 * A(q^4) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (v - u^2)^3 - 4 * u^2 * v^3 * (2*v - u^2) * (2 + v^2 - u^2*v).
%F A226559 a(n) = -A092869(2*n + 1) = A230534(2*n + 1).
%e A226559 G.f. = 1 - x - x^2 + 2*x^3 + x^4 - 2*x^5 - 2*x^6 + 2*x^7 + 4*x^8 - 4*x^9 + ...
%e A226559 G.f. = q^3 - q^7 - q^11 + 2*q^15 + q^19 - 2*q^23 - 2*q^27 + 2*q^31 + 4*q^35 + ...
%t A226559 a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{0, 1, 1, -1, -2, -1, 1, 1}[[ Mod[k, 8] + 1]], {k, n}], {x, 0, n}];
%o A226559 (PARI) {a(n) = if( n<0, 0, polcoeff( prod( k=1, n, (1 - x^k + x * O(x^n))^[ 0, 1, 1, -1, -2, -1, 1, 1][k%8 + 1]), n))};
%Y A226559 Cf. A092869, A230534.
%K A226559 sign
%O A226559 0,4
%A A226559 _Michael Somos_, Jun 10 2013