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A213023 Expansion of psi(x)^2 * psi(-x^3) / chi(-x^2) in powers of x where psi(), chi() are Ramanujan theta functions.

%I A213023 #13 Feb 16 2025 08:33:17
%S A213023 1,2,2,3,2,2,4,4,5,3,4,5,4,6,4,4,5,7,5,3,6,8,8,8,6,3,7,6,10,6,5,10,4,
%T A213023 8,7,8,10,6,9,8,5,10,10,11,6,9,11,6,12,9,8,8,10,9,6,6,15,12,9,9,6,13,
%U A213023 10,13,10,7,14,12,12,8,7,13,10,16,9,10,10,12
%N A213023 Expansion of psi(x)^2 * psi(-x^3) / chi(-x^2) in powers of x where psi(), chi() are Ramanujan theta functions.
%C A213023 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A213023 G. C. Greubel, <a href="/A213023/b213023.txt">Table of n, a(n) for n = 0..1000</a>
%H A213023 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A213023 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A213023 Expansion of q^(-17/24) * eta(q^2)^3 * eta(q^3) * eta(q^4) * eta(q^12) / (eta(q)^2 * eta(q^6)) in powers of q.
%F A213023 Euler transform of period 12 sequence [ 2, -1, 1, -2, 2, -1, 2, -2, 1, -1, 2, -3, ...].
%F A213023 a(n) = A180312(3*n + 1).
%e A213023 1 + 2*x + 2*x^2 + 3*x^3 + 2*x^4 + 2*x^5 + 4*x^6 + 4*x^7 + 5*x^8 + 3*x^9 + ...
%e A213023 q^17 + 2*q^41 + 2*q^65 + 3*q^89 + 2*q^113 + 2*q^137 + 4*q^161 + 4*q^185 + ...
%t A213023 QP := QPochhammer; a[n_]:=SeriesCoefficient[(QP[q^2]^3*QP[q^3]*QP[q^4] *QP[q^12])/(QP[q]^2*QP[q^6]), {q, 0, n}]; Table[a[n], {n,0,50}] (* _G. C. Greubel_, Jan 07 2018 *)
%o A213023 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A) / (eta(x + A)^2 * eta(x^6 + A)), n))}
%Y A213023 Cf. A180312.
%K A213023 nonn
%O A213023 0,2
%A A213023 _Michael Somos_, Jun 03 2012