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A258779 Expansion of (f(-x) * phi(x))^2 in powers of x where phi(), f() are Ramanujan theta functions.

%I A258779 #10 Feb 16 2025 08:33:25
%S A258779 1,2,-5,-10,9,14,-10,0,14,2,-11,-32,0,14,-9,26,2,0,16,-22,14,0,0,26,
%T A258779 -17,-32,-22,-10,-34,14,45,38,0,-34,38,-22,2,0,-10,64,-20,0,0,0,-23,
%U A258779 -46,16,0,-46,-32,26,-10,25,18,0,38,50,0,0,-22,-80,50,0,26,2
%N A258779 Expansion of (f(-x) * phi(x))^2 in powers of x where phi(), f() are Ramanujan theta functions.
%C A258779 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A258779 G. C. Greubel, <a href="/A258779/b258779.txt">Table of n, a(n) for n = 0..2500</a>
%H A258779 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A258779 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A258779 Expansion of q^(-1/12) * (eta(q^2)^5 / (eta(q) * eta(q^4)^2))^2 in powers of q.
%F A258779 Euler transform of period 4 sequence [ 2, -8, 2, -4, ...].
%F A258779 a(n) = A000727(2*n) = A187076(2*n) = A106508(4*n) = A187149(4*n).
%F A258779 Convolution square of A143378.
%e A258779 G.f. = 1 + 2*x - 5*x^2 - 10*x^3 + 9*x^4 + 14*x^5 - 10*x^6 + 14*x^8 + ...
%e A258779 G.f. = q + 2*q^13 - 5*q^25 - 10*q^37 + 9*q^49 + 14*q^61 - 10*q^73 + ...
%t A258779 a[ n_] := SeriesCoefficient[ (QPochhammer[ x] EllipticTheta[ 3, 0, x])^2, {x, 0, n}];
%o A258779 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^5 / (eta(x + A) * eta(x^4 + A)^2))^2, n))};
%Y A258779 Cf. A000727, A106508, A143378, A187076, A187149.
%K A258779 sign
%O A258779 0,2
%A A258779 _Michael Somos_, Jun 09 2015