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A233006 Expansion of psi(x) / f(-x^6) in powers of x where psi(), f() are Ramanujan theta functions.

%I A233006 #14 Feb 16 2025 08:33:20
%S A233006 1,1,0,1,0,0,2,1,0,1,1,0,3,2,0,3,1,0,5,3,0,5,2,0,8,5,0,8,4,0,12,7,0,
%T A233006 12,6,0,19,11,0,19,9,0,27,15,0,28,14,0,39,22,0,41,20,0,55,31,0,58,29,
%U A233006 0,77,43,0,82,41,0,106,58,0,113,57,0,145,80,0,156
%N A233006 Expansion of psi(x) / f(-x^6) in powers of x where psi(), f() are Ramanujan theta functions.
%C A233006 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A233006 G. C. Greubel, <a href="/A233006/b233006.txt">Table of n, a(n) for n = 0..2500</a>
%H A233006 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A233006 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A233006 Expansion of q^(-1/8) * eta(q^2)^2 / (eta(q) * eta(q^6)) in powers of q.
%F A233006 Euler transform of period 6 sequence [ 1, -1, 1, -1, 1, 0, ...].
%F A233006 G.f. is a period 1 Fourier series which satisfies f(-1 / (1152 t)) = (3/2)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A070047.
%F A233006 G.f.: Product_{k>0} (1 + x^k) / (1 + x^(2*k) + x^(4*k)).
%F A233006 a(3*n) = A070047(n). a(3*n + 1) = A097451(n). a(3*n + 2) = 0.
%e A233006 G.f. = 1 + x + x^3 + 2*x^6 + x^7 + x^9 + x^10 + 3*x^12 + 2*x^13 + 3*x^15 + ...
%e A233006 G.f. = q + q^9 + q^25 + 2*q^49 + q^57 + q^73 + q^81 + 3*q^97 + 2*q^105 + ...
%t A233006 a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^(1/2)] / (2 x^(1/8) QPochhammer[ x^6]), {x, 0, n}];
%o A233006 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 / (eta(x + A) * eta(x^6 + A)), n))};
%Y A233006 Cf. A070047, A097451.
%K A233006 nonn
%O A233006 0,7
%A A233006 _Michael Somos_, Dec 03 2013