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A260166 Expansion of phi(x^2) * f(-x^3)^3 / chi(-x)^2 in powers of x where phi(), chi(), f() are Ramanujan theta functions.

%I A260166 #13 Feb 16 2025 08:33:26
%S A260166 1,2,5,7,9,11,10,15,14,19,21,21,28,24,29,26,26,35,37,39,41,34,43,47,
%T A260166 49,56,42,55,57,50,56,50,60,74,69,76,52,70,84,79,81,66,85,74,98,91,74,
%U A260166 88,97,99,86,84,105,107,109,122,90,98,124,119,121,105,125,127
%N A260166 Expansion of phi(x^2) * f(-x^3)^3 / chi(-x)^2 in powers of x where phi(), chi(), f() are Ramanujan theta functions.
%C A260166 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A260166 G. C. Greubel, <a href="/A260166/b260166.txt">Table of n, a(n) for n = 0..1000</a>
%H A260166 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A260166 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A260166 Expansion of q^(-11/24) * eta(q^3)^3 * eta(q^4)^5 / (eta(q)^2 * eta(q^8)^2) in powers of q.
%F A260166 Euler transform of period 24 sequence [ 2, 2, -1, -3, 2, -1, 2, -1, -1, 2, 2, -6, 2, 2, -1, -1, 2, -1, 2, -3, -1, 2, 2, -4, ...].
%F A260166 3 * a(n) = A260158(4*n + 1).
%e A260166 G.f. = 1 + 2*x + 5*x^2 + 7*x^3 + 9*x^4 + 11*x^5 + 10*x^6 + 15*x^7 + 14*x^8 + ...
%e A260166 G.f. = q^11 + 2*q^35 + 5*q^59 + 7*q^83 + 9*q^107 + 11*q^131 + 10*q^155 + ...
%t A260166 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^2] QPochhammer[ x^3]^3 QPochhammer[ -x, x]^2, {x, 0, n}];
%o A260166 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^3 * eta(x^4 + A)^5 / (eta(x + A)^2 * eta(x^8 + A)^2), n))};
%Y A260166 Cf. A260158.
%K A260166 nonn
%O A260166 0,2
%A A260166 _Michael Somos_, Nov 09 2015