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A263021 Expansion of f(-x^3)^6 / (phi(-x) * phi(-x^3)) in powers of x where phi(), f() are Ramanujan theta functions.

%I A263021 #10 Feb 16 2025 08:33:27
%S A263021 1,2,4,4,6,8,9,10,8,14,14,16,16,16,20,18,22,24,21,26,28,28,28,24,36,
%T A263021 34,36,38,32,32,40,42,44,36,46,56,43,50,40,52,54,56,54,42,60,62,64,64,
%U A263021 56,66,56,72,70,56,74,74,76,72,64,80,81,84,84,64,76,88,88
%N A263021 Expansion of f(-x^3)^6 / (phi(-x) * phi(-x^3)) in powers of x where phi(), f() are Ramanujan theta functions.
%C A263021 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A263021 G. C. Greubel, <a href="/A263021/b263021.txt">Table of n, a(n) for n = 0..1000</a>
%H A263021 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A263021 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A263021 Expansion of q^(-3/4) * eta(q^2) * eta(q^3)^4 * eta(q^6) / eta(q)^2 in powers of q.
%F A263021 Euler transform of period 6 sequence [ 2, 1, -2, 1, 2, -4, ...].
%F A263021 a(3*n) = A261445(n). a(3*n + 1) = 2 * A260518(n). a(3*n + 2) = 4 * A260295(n).
%e A263021 G.f. = 1 + 2*x + 4*x^2 + 4*x^3 + 6*x^4 + 8*x^5 + 9*x^6 + 10*x^7 + 8*x^8 + ...
%e A263021 G.f. = q^3 + 2*q^7 + 4*q^11 + 4*q^15 + 6*q^19 + 8*q^23 + 9*q^27 + 10*q^31 + ...
%t A263021 a[ n_] := SeriesCoefficient[ QPochhammer[ x^3]^6 / (EllipticTheta[ 4, 0, x] EllipticTheta[ 4, 0, x^3]), {x, 0, n}];
%o A263021 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^4 * eta(x^6 + A) / eta(x + A)^2, n))};
%Y A263021 Cf. A260295, A260518, A261445.
%K A263021 nonn
%O A263021 0,2
%A A263021 _Michael Somos_, Oct 07 2015