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A230204 Expansion of phi(-x) * f(x^3, x^5) in powers of x where phi(), f() are Ramanujan theta functions.

%I A230204 #11 Feb 16 2025 08:33:20
%S A230204 1,-2,0,1,0,1,-2,2,0,0,0,0,-2,0,-1,-2,2,0,3,0,0,2,2,-2,0,-2,0,-2,-2,0,
%T A230204 0,0,0,1,0,0,2,2,0,1,-2,2,-2,0,0,0,0,0,-2,0,2,0,-2,0,0,2,0,0,-2,0,1,
%U A230204 -2,0,-2,2,0,0,0,1,0,0,0,2,0,0,2,2,-2,2,0,0
%N A230204 Expansion of phi(-x) * f(x^3, x^5) in powers of x where phi(), f() are Ramanujan theta functions.
%C A230204 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A230204 G. C. Greubel, <a href="/A230204/b230204.txt">Table of n, a(n) for n = 0..1000</a>
%H A230204 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A230204 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A230204 Euler transform of period 16 sequence [ -2, -1, -1, -1, -1, -2, -2, -2, -2, -2, -1, -1, -1, -1, -2, -2, ...].
%F A230204 a(n) = A030204(2*n).
%e A230204 G.f. = 1 - 2*x + x^3 + x^5 - 2*x^6 + 2*x^7 - 2*x^12 - x^14 - 2*x^15 + ...
%e A230204 G.f. = q - 2*q^17 + q^49 + q^81 - 2*q^97 + 2*q^113 - 2*q^193 - q^225 + ...
%t A230204 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] QPochhammer[ -q^3, q^8] QPochhammer[ -q^5, q^8] QPochhammer[ q^8], {q, 0, n}];
%o A230204 (PARI) {a(n) = local(m, j); if( n<0, 0, m = 16*n + 1; sum( k=0, sqrtint(m \ 4), if( issquare(m - 16*k^2, &j), if( k==0, 1, 2) * (-1)^k * ((j%8)==1 || (j%8==7)))))}
%Y A230204 Cf. A030204.
%K A230204 sign
%O A230204 0,2
%A A230204 _Michael Somos_, Oct 11 2013