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%I #16 Mar 12 2021 22:24:45
%S 1,2,5,8,16,26,45,70,113,170,261,382,567,812,1171,1646,2322,3212,4448,
%T 6066,8272,11142,14992,19970,26561,35032,46117,60280,78631,101946,
%U 131888,169724,217937,278548,355237,451178,571799,722002,909744,1142502,1431889
%N Expansion of phi(-q^3) / f(-q)^2 in powers of q where phi(), f() are Ramanujan theta functions.
%H G. C. Greubel, <a href="/A137685/b137685.txt">Table of n, a(n) for n = 0..1000</a>
%H G. E. Andrews, <a href="http://dx.doi.org/10.1090/cbms/066">q-series</a>, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 71, Equ. (7.30). MR0858826 (88b:11063).
%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 6 sequence [ 2, 2, 0, 2, 2, 1, ...].
%F G.f.: Product_{k>0} (1 + x^k + x^(2*k)) / ( (1 - x^(2*k)) * (1 - x^k +x^(2*k)) ).
%e G.f. = 1 + 2*q + 5*q^2 + 8*q^3 + 16*q^4 + 26*q^5 + 45*q^6 + 70*q^7 + 113*q^8 + ...
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^3] / QPochhammer[ q]^2, {q, 0, n}]; (* _Michael Somos_, Oct 04 2015 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( sum(k=1, sqrtint(n \ 3), 2 * (-1)^k * x^(3*k^2), 1 + A) / eta(x + A)^2, n))};
%o (PARI) {a(N) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^2 / (eta(x + A)^2 * eta(x^6 + A)), n))}; /* _Michael Somos_, Oct 04 2015 */
%o (PARI) q='q+O('q^99); Vec(eta(q^3)^2/(eta(q)^2*eta(q^6))) \\ _Altug Alkan_, Mar 30 2018
%Y Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%K nonn
%O 0,2
%A _Michael Somos_, Feb 05 2008
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