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%I #18 Mar 12 2021 22:24:44
%S 1,6,9,-16,-66,-54,98,300,243,-364,-1128,-828,1221,3498,2511,-3528,
%T -9876,-6804,9358,25428,17217,-23068,-61644,-40824,53916,141318,92340,
%U -119912,-310554,-199980,256792,656436,418311,-530960,-1344144,-847584,1066157,2673372
%N Expansion of (f(x) / f(x^3))^6 in powers of x where f() is a Ramanujan theta function.
%C theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A132107/b132107.txt">Table of n, a(n) for n = 0..1000</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Euler transform of period 12 sequence [ 6, -12, 0, -6, 6, 0, 6, -6, 0, -12, 6, 0, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 27 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A264026.
%F G.f.: (Product_{k>0} (1 + x^k) * (1 - x^(2*k)) * (1 + x^(3*k)) * (1 - x^(6*k)) / ((1 + x^(2*k)) * (1 + x^(6*k))))^6.
%e G.f. = 1 + 6*x + 9*x^2 - 16*x^3 - 66*x^4 - 54*x^5 + 98*x^6 + 300*x^7 + 243*x^8 + ...
%e G.f. = 1/q + 6*q + 9*q^3 - 16*q^5 - 66*q^7 - 54*q^9 + 98*q^11 + 300*q^13 + ...
%t a[ n_] := SeriesCoefficient[ (QPochhammer[ -x] / QPochhammer[ -x^3])^6, {x, 0, n}]; (* _Michael Somos_, Aug 26 2015 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^3))^6, n))};
%Y Cf. A007262, A264026.
%K sign
%O 0,2
%A _Michael Somos_, Aug 09 2007
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