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A143161 Expansion of chi(-x)^2 * chi(-x^2) in powers of x where chi() is a Ramanujan theta function.

%I A143161 #13 Feb 16 2025 08:33:08
%S A143161 1,-2,0,0,3,-2,0,0,4,-6,0,0,7,-8,0,0,13,-14,0,0,19,-20,0,0,29,-34,0,0,
%T A143161 43,-46,0,0,62,-70,0,0,90,-96,0,0,126,-138,0,0,174,-186,0,0,239,-262,
%U A143161 0,0,325,-346,0,0,435,-472,0,0,580,-620,0,0,769,-826,0,0
%N A143161 Expansion of chi(-x)^2 * chi(-x^2) in powers of x where chi() is a Ramanujan theta function.
%C A143161 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A143161 G. C. Greubel, <a href="/A143161/b143161.txt">Table of n, a(n) for n = 0..1000</a>
%H A143161 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A143161 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A143161 Expansion of q^(1/6) * eta(q)^2 / (eta(q^2) * eta(q^4)) in powers of q.
%F A143161 Euler transform of period 4 sequence [ -2, -1, -2, 0, ...].
%F A143161 G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 8^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A098613. - _Michael Somos_, Sep 07 2015
%F A143161 a(4*n + 2) = a(4*n + 3) = 0.
%F A143161 G.f.: (Product_{k>0} (1 + x^k)^2 * (1 + x^(2*k)))^-1.
%F A143161 a(4*n) = A029552(n). a(4*n + 1) = -2 * A098613(n).
%e A143161 G.f. = 1 - 2*x + 3*x^4 - 2*x^5 + 4*x^8 - 6*x^9 + 7*x^12 - 8*x^13 + 13*x^16 + ...
%e A143161 G.f. = 1/q - 2*q^5 + 3*q^23 - 2*q^29 + 4*q^47 - 6*q^53 + 7*q^71 - 8*q^77 + ...
%t A143161 a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2]^2 QPochhammer[ x^2, x^4], {x, 0, n}]; (* _Michael Somos_, Sep 07 2015 *)
%o A143161 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 / (eta(x^2 + A) * eta(x^4 + A)), n))};
%Y A143161 Cf. A029552, A098613.
%K A143161 sign
%O A143161 0,2
%A A143161 _Michael Somos_, Jul 27 2008