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A258278 Expansion of f(-x, -x^5)^2 in powers of x where f(,) is Ramanujan's general theta function.

%I A258278 #11 Feb 16 2025 08:33:25
%S A258278 1,-2,1,0,0,-2,2,0,2,-2,1,0,0,-2,0,0,3,-2,0,0,0,-4,2,0,2,0,2,0,0,-2,0,
%T A258278 0,1,-2,2,0,0,-2,2,0,2,-4,1,0,0,-2,0,0,2,-2,0,0,0,0,2,0,4,-2,0,0,0,-4,
%U A258278 0,0,2,-2,3,0,0,0,2,0,2,-4,0,0,0,-2,0,0,1
%N A258278 Expansion of f(-x, -x^5)^2 in powers of x where f(,) is Ramanujan's general theta function.
%C A258278 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A258278 G. C. Greubel, <a href="/A258278/b258278.txt">Table of n, a(n) for n = 0..1000</a>
%H A258278 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A258278 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A258278 Expansion of (chi(-x) * psi(x^3))^2 in powers of x where chi(), psi() are Ramanujan theta functions.
%F A258278 Expansion of q^(-2/3) * (eta(q) * eta(q^6)^2 / (eta(q^2) * eta(q^3)))^2 in powers of q.
%F A258278 Euler transform of period 6 sequence [ -2, 0, 0, 0, -2, -2, ...].
%F A258278 a(n) = (-1)^n * A122856(n). Convolution square of A089802.
%F A258278 a(2*n) = A122865(n). a(4*n +1) = -2 * A121444(n). a(4*n + 2) = A122856(n). a(4*n + 3) = a(8*n + 4) = 0. a(8*n) = A002175(n). a(8*n + 2) = A122856(n).
%F A258278 2 * a(n) = -A258210(3*n + 2). - _Michael Somos_, May 01 2016
%e A258278 G.f. = 1 - 2*x + x^2 - 2*x^5 + 2*x^6 + 2*x^8 - 2*x^9 + x^10 - 2*x^13 + ...
%e A258278 G.f. = q^2 - 2*q^5 + q^8 - 2*q^17 + 2*q^20 + 2*q^26 - 2*q^29 + q^32 + ...
%t A258278 a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2]^2 EllipticTheta[ 2, 0, x^(3/2)]^2 / (4 x^(3/4)), {x, 0, n}];
%o A258278 (PARI) {a(n) = if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^6 + A)^2 / (eta(x^2 + A) * eta(x^3 + A)))^2, n))};
%Y A258278 Cf. A002175, A089802, A121444, A122856, A122865, A258210.
%K A258278 sign
%O A258278 0,2
%A A258278 _Michael Somos_, May 25 2015