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A185220 Expansion of phi(x^3) * psi(x)^2 / chi(-x) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

%I A185220 #19 Feb 16 2025 08:33:13
%S A185220 1,3,4,5,5,5,7,7,9,7,6,11,8,10,8,9,14,10,15,7,7,14,14,16,8,13,13,12,
%T A185220 18,14,13,15,15,16,9,11,22,16,19,16,11,17,16,23,19,9,22,18,16,15,18,
%U A185220 27,12,23,11,15,24,24,27,9,23,23,20,21,19,15,22,24,22,17
%N A185220 Expansion of phi(x^3) * psi(x)^2 / chi(-x) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.
%C A185220 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A185220 Vaclav Kotesovec, <a href="/A185220/b185220.txt">Table of n, a(n) for n = 0..1000</a>
%H A185220 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A185220 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A185220 Expansion of q^(-7/24) * eta(q^2)^5 * eta(q^3)^2 / (eta(q)^3 * eta(q^6)) in powers of q.
%F A185220 Euler transform of period 6 sequence [ 3, -2, 1, -2, 3, -3, ...].
%F A185220 G.f.: Product_{k>0} (1 - x^k)^2 * (1 + x^k)^5 * (1 - x^(3*k)) / (1 + x^(3*k)).
%F A185220 a(n) = A224825(3*n) = A227595(3*n).
%e A185220 1 + 3*x + 4*x^2 + 5*x^3 + 5*x^4 + 5*x^5 + 7*x^6 + 7*x^7 + 9*x^8 + 7*x^9 + ...
%e A185220 q^7 + 3*q^31 + 4*q^55 + 5*q^79 + 5*q^103 + 5*q^127 + 7*q^151 + 7*q^175 + ...
%t A185220 nmax = 100; CoefficientList[Series[Product[(1 - x^k)^2 * (1 + x^k)^5 * (1 - x^(3*k)) / (1 + x^(3*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 08 2015 *)
%o A185220 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A)^2 / (eta(x + A)^3 * eta(x^6 + A)), n))}
%Y A185220 Cf. A224825, A227595.
%K A185220 nonn
%O A185220 0,2
%A A185220 _Michael Somos_, Aug 29 2013