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A226556 Expansion of f(x, -x^4) / f(-x^2, x^3) in powers of x where f(,) is Ramanujan's general theta function.

%I A226556 #21 Feb 16 2025 08:33:20
%S A226556 1,1,1,0,-1,-1,-1,-1,0,1,2,3,2,0,-2,-4,-4,-3,-1,3,6,7,5,0,-5,-9,-10,
%T A226556 -7,-1,7,14,16,11,1,-11,-20,-22,-16,-2,15,29,33,23,2,-23,-41,-45,-32,
%U A226556 -4,30,57,64,45,4,-43,-78,-86,-60,-7,57,107,119,83,8,-79
%N A226556 Expansion of f(x, -x^4) / f(-x^2, x^3) in powers of x where f(,) is Ramanujan's general theta function.
%C A226556 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A226556 Given the g.f. A(x), then S(q) := q^(1/5) * A(q) notation is used by Berndt.
%D A226556 G. E. Andrews and B. C. Berndt, Ramanujan's Lost Notebook, Part I, Springer, 2005, see p. 57.
%D A226556 B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 9.
%H A226556 Seiichi Manyama, <a href="/A226556/b226556.txt">Table of n, a(n) for n = 0..10000</a>
%H A226556 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A226556 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A226556 O. Marichev and M. Trott, <a href="http://blog.wolfram.com/2013/05/01/after-100-years-ramanujan-gap-filled/">After 100 Years, Ramanujan Gap Filled</a>
%F A226556 Given g.f. A(x), then B(x) = x * A(x^5) satisfies 0 = f(B(x), B(x^3)) where f(u, v) = (u^3 - v) * (1 + u*v^3) - 3 * u^2*v^2.
%F A226556 Given g.f. A(x), then B(x) = x * A(x^5) satisfies 0 = f(B(x), B(x^5)) where f(u, v) = u^5 * (1 - 3*v + 4*v^2 - 2*v^3 + v^4) - v * (1 + 2*v + 4*v^2 + 3*v^3 + v^4).
%F A226556 Euler transform of period 20 sequence [1, 0, -1, -1, 0, 0, -1, 1, 1, 0, 1, 1, -1, 0, 0, -1, -1, 0, 1, 0, ...].
%F A226556 G.f.: (Sum_{k in Z} (-1)^k * (-x)^((5*k + 3)*k / 2)) / (Sum_{k in Z} (-1)^k * (-x)^((5*k + 1)*k / 2)).
%F A226556 G.f.: 1 / (1 - x / (1 + x^2 / (1 - x^3 / ...))). [continued fraction]
%F A226556 a(n) = (-1)^n * A007325(n).
%F A226556 G.f.: 1/G(0), where G(k)= 1 + (-x)^(k+1)/G(k+1) ; (continued fraction). - _Sergei N. Gladkovskii_, Jun 29 2013
%e A226556 G.f. = 1 - x + x^2 - x^4 + x^5 - x^6 + x^7 - x^9 + 2*x^10 - 3*x^11 + 2*x^12 + ...
%e A226556 G.f. = q - q^6 + q^11 - q^21 + q^26 - q^31 + q^36 - q^46 + 2*q^51 - 3*q^56 + 2*q^61 + ...
%t A226556 a[ n_] := SeriesCoefficient[ ContinuedFractionK[ (-q)^k, 1, {k, 0, n}], {q, 0, n}];
%t A226556 a[ n_] := SeriesCoefficient[ QPochhammer[ -q, -q^5] QPochhammer[ q^4, -q^5] / (QPochhammer[ q^2, -q^5] QPochhammer[-q^3, -q^5]), {q, 0, n}];
%o A226556 (PARI) {a(n) = my(k); if( n<0, 0, k = (3 + sqrtint(9 + 40*n)) \ 10; polcoeff( sum( n=-k, k, (-1)^n * (-x)^( (5*n^2 + 3*n) / 2), x * O(x^n)) / sum( n=-k, k, (-1)^n * (-x)^( (5*n^2 + n) / 2), x * O(x^n)), n))};
%Y A226556 Cf. A007325.
%K A226556 sign
%O A226556 0,11
%A A226556 _Michael Somos_, Jun 10 2013