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A263051 Expansion of f(-x) * f(x^2, x^10) / f(-x^3)^2 in powers of x where f(, ) is Ramanujan's general theta function.

%I A263051 #13 Feb 16 2025 08:33:27
%S A263051 1,-1,0,1,-3,1,3,-5,2,6,-10,4,10,-18,7,17,-30,12,28,-49,19,44,-78,31,
%T A263051 69,-120,47,105,-182,71,156,-271,106,229,-396,154,333,-572,222,475,
%U A263051 -817,317,673,-1151,445,943,-1608,620,1307,-2226,857,1798,-3053,1173,2455
%N A263051 Expansion of f(-x) * f(x^2, x^10) / f(-x^3)^2 in powers of x where f(, ) is Ramanujan's general theta function.
%C A263051 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A263051 G. C. Greubel, <a href="/A263051/b263051.txt">Table of n, a(n) for n = 0..2500</a>
%H A263051 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A263051 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A263051 Expansion of q^(-11/24) * eta(q) * eta(q^4)^2 * eta(q^6) * eta(q^24) / (eta(q^2) * eta(q^3)^2 * eta(q^8) * eta(q^12)) in powers of q.
%F A263051 Euler transform of period 24 sequence [-1, 0, 1, -2, -1, 1, -1, -1, 1, 0, -1, 0, -1, 0, 1, -1, -1, 1, -1, -2, 1, 0, -1, 0, ...].
%F A263051 a(n) = - A137569(2*n + 1).
%e A263051 G.f. = 1 - x + x^3 - 3*x^4 + x^5 + 3*x^6 - 5*x^7 + 2*x^8 + 6*x^9 + ...
%e A263051 G.f. = q^11 - q^35 + q^83 - 3*q^107 + q^131 + 3*q^155 - 5*q^179 + ...
%t A263051 a[ n_] := SeriesCoefficient[ 2^(-1/2) x^(-3/4) EllipticTheta[ 2, Pi/4, x^3] QPochhammer[ -x^2, x^4] QPochhammer[ x] / QPochhammer[ x^3]^2, {x, 0, n}];
%o A263051 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A)^2 * eta(x^6 + A) * eta(x^24 + A) / (eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^8 + A) * eta(x^12 + A)), n))};
%o A263051 (PARI) q='q+O('q^99); Vec(eta(q)*eta(q^4)^2*eta(q^6)*eta(q^24)/(eta(q^2)*eta(q^3)^2*eta(q^8)*eta(q^12))) \\ _Altug Alkan_, Jul 31 2018
%Y A263051 Cf. A137569.
%K A263051 sign
%O A263051 0,5
%A A263051 _Michael Somos_, Oct 08 2015