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A227398 Expansion of chi(x^3) / chi(x) in powers of x where chi() is a Ramanujan theta function.

%I A227398 #19 Feb 16 2025 08:33:20
%S A227398 1,-1,1,-1,1,-2,2,-3,3,-3,4,-5,6,-7,8,-9,10,-12,14,-16,18,-20,23,-26,
%T A227398 30,-34,38,-42,47,-53,60,-67,74,-82,91,-102,114,-126,139,-153,169,
%U A227398 -187,207,-228,250,-274,301,-331,364,-399,436,-476,520,-569,622,-679
%N A227398 Expansion of chi(x^3) / chi(x) in powers of x where chi() is a Ramanujan theta function.
%C A227398 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A227398 G. C. Greubel, <a href="/A227398/b227398.txt">Table of n, a(n) for n = 0..2500</a>
%H A227398 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A227398 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A227398 Expansion of f(-x^6) / f(x, x^5) in powers of x where f(,) is Ramanujan's general theta function.
%F A227398 Expansion of q^(1/12) * eta(q) * eta(q^4) * eta(q^6)^2 / (eta(q^2)^2 * eta(q^3) * eta(q^12)) in powers of q.
%F A227398 Euler transform of period 12 sequence [-1, 1, 0, 0, -1, 0, -1, 0, 0, 1, -1, 0, ...].
%F A227398 G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. for A098884.
%F A227398 G.f.: 1 / Product_{k>0} (1 - (-x)^k + x^(2*k)).
%F A227398 G.f.: Product_{k>0} (1 - x^(2*k - 1) + x^(4*k - 2)).
%F A227398 G.f.: Product_{k>0} ((1 + x^(6*k - 3)) / (1 + x^(2*k - 1))).
%F A227398 G.f.: 1 / Product_{k>0} ((1 + x^(6*k - 1)) * (1 + x^(6*k - 5))).
%F A227398 G.f.: Product_{k>0} (1  + (-x)^(3*k - 1)) * (1 + (-x)^(3*k - 2)).
%F A227398 G.f.: (Sum_{k in Z} (-1)^k * x^(3*k * (3*k-1))) / (Sum_{k in Z} x^(k * (3*k - 2))).
%F A227398 a(n) = (-1)^n * A003105(n). Convolution inverse of A098884.
%e A227398 G.f. = 1 - x + x^2 - x^3 + x^4 - 2*x^5 + 2*x^6 - 3*x^7 + 3*x^8 - 3*x^9 + ...
%e A227398 G.f. = 1/q - q^11 + q^23 - q^35 + q^47 - 2*q^59 + 2*q^71 - 3*q^83 + 3*q^95 + ...
%t A227398 a[ n_] := SeriesCoefficient[ 1 / Product[ 1 - (-x)^k + x^(2 k), {k, n}], {x, 0, n}];
%t A227398 a[ n_] := SeriesCoefficient[ Product[ 1 - x^k + x^(2 k), {k, 1, n, 2}], {x, 0, n}];
%t A227398 a[ n_] := SeriesCoefficient[ Product[ 1 + x^k, {k, 3, n, 6}] / Product[ 1 + x^k, {k, 1, n, 2}], {x, 0, n}];
%t A227398 a[ n_] := SeriesCoefficient[ 1 / (Product[ 1 + x^k, {k, 1, n, 6}] Product[ 1 + x^k, {k, 5, n, 6}]), {x, 0, n}];
%t A227398 a[ n_] := SeriesCoefficient[ Product[ 1 + (-x)^k, {k, 1, n, 3}] Product[ 1 + (-x)^k, {k, 2, n, 3}], {x, 0, n}];
%t A227398 a[ n_] := SeriesCoefficient[ QPochhammer[ -x^3, x^6] / QPochhammer[ -x, x^2], {x, 0, n}];
%o A227398 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^2 / (eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^12 + A)), n))};
%o A227398 (PARI) {a(n) = my(A, m); if( n<0, 0, A = x * O(x^n); m = sqrtint(3*n + 1); polcoeff( eta(x^6 + A) / sum(k= -((m-1)\3), (m+1)\3, x^(k * (3*k - 2)), A), n))};
%Y A227398 Cf. A003105, A098884.
%K A227398 sign
%O A227398 0,6
%A A227398 _Michael Somos_, Sep 20 2013