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A260941 Expansion of phi(-x) * phi(x^6) / chi(-x^3) in powers of x where phi(), chi() are Ramanujan theta functions.

%I A260941 #11 Feb 16 2025 08:33:26
%S A260941 1,-2,0,1,0,0,3,-4,0,2,-2,0,2,0,0,1,-4,0,0,0,0,3,0,0,2,-4,0,4,-2,0,2,
%T A260941 0,0,0,-8,0,1,0,0,2,0,0,2,0,0,3,0,0,0,0,0,2,-4,0,2,-6,0,2,0,0,4,-4,0,
%U A260941 0,-4,0,1,0,0,4,0,0,2,0,0,2,0,0,1,-4,0,0,-4
%N A260941 Expansion of phi(-x) * phi(x^6) / chi(-x^3) in powers of x where phi(), chi() are Ramanujan theta functions.
%C A260941 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A260941 G. C. Greubel, <a href="/A260941/b260941.txt">Table of n, a(n) for n = 0..2500</a>
%H A260941 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A260941 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A260941 Expansion of q^(-1/8) * eta(q)^2 * eta(q^12)^5 / (eta(q^2) * eta(q^3) * eta(q^6) * eta(q^24)^2) in powers of q.
%F A260941 Euler transform of period 24 sequence [ -2, -1, -1, -1, -2, 1, -2, -1, -1, -1, -2, -4, -2, -1, -1, -1, -2, 1, -2, -1, -1, -1, -2, -2, ...].
%F A260941 a(3*n) = A131961(n). a(3*n + 1) = -2 * A112608(n). a(3*n + 2) = 0.
%e A260941 G.f. = 1 - 2*x + x^3 + 3*x^6 - 4*x^7 + 2*x^9 - 2*x^10 + 2*x^12 + x^15 + ...
%e A260941 G.f. = q - 2*q^9 + q^25 + 3*q^49 - 4*q^57 + 2*q^73 - 2*q^81 + 2*q^97 + ...
%t A260941 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] EllipticTheta[ 3, 0, x^6] QPochhammer[ -x^3, x^3], {x, 0, n}];
%o A260941 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^12 + A)^5 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^6 + A) * eta(x^24 + A)^2), n))};
%o A260941 (PARI) q='q+O('q^99); Vec(eta(q)^2*eta(q^12)^5/(eta(q^2)*eta(q^3)*eta(q^6)*eta(q^24)^2)) \\ _Altug Alkan_, Aug 01 2018
%Y A260941 Cf. A112608, A131961.
%K A260941 sign
%O A260941 0,2
%A A260941 _Michael Somos_, Aug 04 2015