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A257873 Expansion of f(-x)^2 * chi(-x^4) * psi(x^6) in powers of x where psi(), chi(), f() are Ramanujan theta functions.

%I A257873 #10 Feb 16 2025 08:33:25
%S A257873 1,-2,-1,2,0,4,0,-4,-4,-2,3,4,0,4,0,-8,5,-6,0,6,0,4,0,-4,-4,-8,-4,10,
%T A257873 0,8,0,-4,9,-6,-4,6,0,8,0,-8,-12,-12,3,6,0,12,0,-12,8,-6,12,8,0,8,0,
%U A257873 -12,-8,-10,-4,6,0,12,0,-8,8,-10,-5,16,0,8,0,-12,-12
%N A257873 Expansion of f(-x)^2 * chi(-x^4) * psi(x^6) in powers of x where psi(), chi(), f() are Ramanujan theta functions.
%C A257873 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A257873 G. C. Greubel, <a href="/A257873/b257873.txt">Table of n, a(n) for n = 0..2500</a>
%H A257873 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A257873 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A257873 Expansion of phi(-x) * phi(-x^4) * f(-x^2, -x^10) in powers of x where phi(), f() are Ramanujan theta functions.
%F A257873 Expansion of q^(-2/3) * eta(q)^2 * eta(q^4) * eta(q^12)^2 / (eta(q^6) * eta(q^8)) in powers of q.
%F A257873 2 * a(n) = A127786(3*n + 2). a(8*n + 4) = a(8*n + 6) = 0.
%e A257873 G.f. = 1 - 2*x - x^2 + 2*x^3 + 4*x^5 - 4*x^7 - 4*x^8 - 2*x^9 + 3*x^10 + ...
%e A257873 G.f. = q^2 - 2*q^5 - q^8 + 2*q^11 + 4*q^17 - 4*q^23 - 4*q^26 - 2*q^29 + ...
%t A257873 a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2 EllipticTheta[ 2, 0, x^3] / (2 x^(3/4) QPochhammer[ -x^4, x^4]), {x, 0, n}];
%o A257873 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A) * eta(x^12 + A)^2 / (eta(x^6 + A) * eta(x^8 + A)), n))};
%Y A257873 Cf. A127786.
%K A257873 sign
%O A257873 0,2
%A A257873 _Michael Somos_, May 11 2015