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A262090 Expansion of f(x^3, x^21) / f(-x^2, -x^4) where f(, ) is the Ramanujan general theta function.

%I A262090 #10 Feb 16 2025 08:33:27
%S A262090 1,0,1,1,2,1,3,2,5,3,7,5,11,7,15,11,22,15,30,22,42,31,56,43,77,58,101,
%T A262090 80,135,106,177,142,232,187,299,246,388,319,495,415,634,532,803,683,
%U A262090 1017,869,1277,1103,1605,1390,2000,1751,2492,2189,3087,2733,3819
%N A262090 Expansion of f(x^3, x^21) / f(-x^2, -x^4) where f(, ) is the Ramanujan general theta function.
%C A262090 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A262090 G. C. Greubel, <a href="/A262090/b262090.txt">Table of n, a(n) for n = 0..2500</a>
%H A262090 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A262090 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A262090 Euler transform of period 48 sequence [ 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, ...].
%F A262090 a(n) = - A143067(2*n + 3).
%e A262090 G.f. = 1 + x^2 + x^3 + 2*x^4 + x^5 + 3*x^6 + 2*x^7 + 5*x^8 + 3*x^9 + ...
%e A262090 G.f. = q^77 + q^173 + q^221 + 2*q^269 + q^317 + 3*q^365 + 2*q^413 + ...
%t A262090 a[ n_] := SeriesCoefficient[ QPochhammer[ -x^3, x^24] QPochhammer[ -x^21, x^24] QPochhammer[ x^24] / QPochhammer[ x^2], {x, 0, n}];
%o A262090 (PARI) {a(n) = if( n<0, 0, A = x * O(x^n); polcoeff( subst( prod(k=1, n\3, 1 - x^k * [1, 1, 0, 0, 0, 0, 0, 1][k%8 + 1], 1 + x * O(x^(n\3))), x, -x^3) / eta(x^2 + x * O(x^n)), n))};
%Y A262090 Cf. A143067.
%K A262090 nonn
%O A262090 0,5
%A A262090 _Michael Somos_, Sep 10 2015