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A181648 Expansion of x^(-2/3) * psi(x) * c(x^2) / 3 in powers of x where psi() is a Ramanujan theta function and c() is a cubic AGM theta function.

%I A181648 #25 Feb 16 2025 08:33:13
%S A181648 1,1,1,2,2,3,1,2,3,2,4,3,3,3,4,3,2,2,6,5,3,5,3,5,4,5,3,4,5,4,5,4,5,7,
%T A181648 6,7,3,3,7,4,8,4,4,5,7,6,5,6,7,8,6,4,6,9,6,8,6,4,4,4,11,7,4,11,4,9,6,
%U A181648 7,8,7,11,5,5,8,8,10,6,5,10,6,8,6,7,7,8
%N A181648 Expansion of x^(-2/3) * psi(x) * c(x^2) / 3 in powers of x where psi() is a Ramanujan theta function and c() is a cubic AGM theta function.
%C A181648 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A181648 G. C. Greubel, <a href="/A181648/b181648.txt">Table of n, a(n) for n = 0..1000</a>
%H A181648 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A181648 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A181648 Expansion of q^(-19/24) * eta(q^2) * eta(q^6)^3 / eta(q) in powers of q.
%F A181648 Euler transform of period 6 sequence [ 1, 0, 1, 0, 1, -3, ...].
%F A181648 3 * a(n) = A008443(3*n + 2).
%e A181648 1 + x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + x^6 + 2*x^7 + 3*x^8 + 2*x^9 + 4*x^10 + ...
%e A181648 q^19 + q^43 + q^67 + 2*q^91 + 2*q^115 + 3*q^139 + q^163 + 2*q^187 + 3*q^211 + ...
%t A181648 A181648[n_]:= SeriesCoefficient[QPochhammer[q^2, q^2]*QPochhammer[q^6, q^6]^3/QPochhammer[q, q], {q, 0, n}]; Table[A181648[n], {n, 0, 50}] (* _G. C. Greubel_, Dec 24 2017 *)
%o A181648 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^+2 + A) * eta(x^6 + A)^3 / eta(x + A), n))}
%Y A181648 Cf. A008443.
%K A181648 nonn
%O A181648 0,4
%A A181648 _Michael Somos_, Jun 22 2012