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A260943 Expansion of psi(-x^2) * chi(x^3) * f(x^6) in powers of x where psi(), chi(), f() are Ramanujan theta functions.

%I A260943 #12 Feb 16 2025 08:33:26
%S A260943 1,0,-1,1,0,-1,0,0,-1,1,0,-2,0,0,0,0,0,-1,3,0,-1,2,0,0,0,0,-1,2,0,0,1,
%T A260943 0,-2,0,0,-1,2,0,-1,0,0,-1,0,0,0,1,0,-1,2,0,-2,0,0,-2,0,0,0,0,0,-1,0,
%U A260943 0,-1,3,0,-1,0,0,-1,0,0,-1,2,0,0,2,0,-1,0,0
%N A260943 Expansion of psi(-x^2) * chi(x^3) * f(x^6) in powers of x where psi(), chi(), f() are Ramanujan theta functions.
%C A260943 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700)
%H A260943 G. C. Greubel, <a href="/A260943/b260943.txt">Table of n, a(n) for n = 0..2500</a>
%H A260943 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A260943 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A260943 Expansion of q^(-3/8) * eta(q^2) * eta(q^6) * eta(q^8) * eta(q^12)^2 / (eta(q^3) * eta(q^4) * eta(q^24)) in powers of q.
%F A260943 Euler transform of period 24 sequence [ 0, -1, 1, 0, 0, -1, 0, -1, 1, -1, 0, -2, 0, -1, 1, -1, 0, -1, 0, 0, 1, -1, 0, -2, ...].
%F A260943 a(3*n) = A112606(n). a(3*n + 1) = 0. a(3*n + 2) = - A131964(n).
%e A260943 G.f. = 1 - x^2 + x^3 - x^5 - x^8 + x^9 - 2*x^11 - x^17 + 3*x^18 - x^20 + ...
%e A260943 G.f. = q^3 - q^19 + q^27 - q^43 - q^67 + q^75 - 2*q^91 - q^139 + 3*q^147 + ...
%t A260943 a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, x] QPochhammer[ -x^6] QPochhammer[ -x^3, x^6] / (2^(1/2) x^(1/4)), {x, 0, n}];
%o A260943 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^6 + A) * eta(x^8 + A) * eta(x^12 + A)^2 / (eta(x^3 + A) * eta(x^4 + A) * eta(x^24 + A)), n))};
%Y A260943 Cf. A112606, A131964.
%K A260943 sign
%O A260943 0,12
%A A260943 _Michael Somos_, Aug 04 2015