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A260183 Expansion of f(x, x^2) * f(x^4, x^8) / f(-x^3, -x^6)^2 in powers of x where f(, ) is Ramanujan's general theta function.

%I A260183 #39 Feb 16 2025 08:33:26
%S A260183 1,1,1,2,3,4,6,8,10,14,18,23,30,38,47,60,74,91,114,139,169,207,250,
%T A260183 301,364,436,520,622,739,875,1038,1224,1439,1694,1985,2321,2714,3162,
%U A260183 3677,4275,4956,5735,6634,7655,8819,10155,11669,13389,15354,17575,20091
%N A260183 Expansion of f(x, x^2) * f(x^4, x^8) / f(-x^3, -x^6)^2 in powers of x where f(, ) is Ramanujan's general theta function.
%C A260183 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A260183 G. C. Greubel, <a href="/A260183/b260183.txt">Table of n, a(n) for n = 0..1000</a>
%H A260183 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A260183 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A260183 Expansion of q^(1/2) * eta(q^2) * eta(q^8) * eta(q^12)^2 / (eta(q) * eta(q^4) * eta(q^6) * eta(q^24)) in powers of q.
%F A260183 Euler transform of period 24 sequence [ 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, ...].
%F A260183 a(n) = A003105(2*n + 1).
%e A260183 G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 10*x^8 + 14*x^9 + ...
%e A260183 G.f. = 1/q + q + q^3 + 2*q^5 + 3*q^7 + 4*q^9 + 6*q^11 + 8*q^13 + 10*q^15 + ...
%t A260183 a[ n_] := SeriesCoefficient[ QPochhammer[ x^2, x^12] QPochhammer[ x^10, x^12] QPochhammer[ x^12, x^24] QPochhammer[ x^8] / QPochhammer[x], {x, 0, n}];
%o A260183 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^8 + A) * eta(x^12 + A)^2 / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^24 + A)), n))};
%Y A260183 Cf. A003105.
%K A260183 nonn
%O A260183 0,4
%A A260183 _Michael Somos_, Nov 10 2015