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A230205 Expansion of phi(-x) * f(x^1, x^7) in powers of x where phi(), f() are Ramanujan theta functions.

%I A230205 #11 Feb 16 2025 08:33:20
%S A230205 1,-1,-2,0,2,2,0,1,-2,-2,-1,0,0,0,2,0,0,2,0,-2,0,0,1,0,0,-2,2,1,-2,0,
%T A230205 0,0,-2,0,0,-2,0,2,2,0,0,0,0,4,0,1,0,-2,0,0,-2,0,-1,-2,-2,0,0,0,2,-2,
%U A230205 0,0,0,2,2,2,0,0,2,0,-2,0,0,0,2,0,-1,-4,0,0
%N A230205 Expansion of phi(-x) * f(x^1, x^7) in powers of x where phi(), f() are Ramanujan theta functions.
%C A230205 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A230205 G. C. Greubel, <a href="/A230205/b230205.txt">Table of n, a(n) for n = 0..1000</a>
%H A230205 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A230205 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A230205 Euler transform of period 16 sequence [ -1, -2, -2, -1, -2, -1, -1, -2, -1, -1, -2, -1, -2, -2, -1, -2, ...].
%F A230205 a(n) = - A030204(2*n + 1).
%e A230205 G.f. = 1 - x - 2*x^2 + 2*x^4 + 2*x^5 + x^7 - 2*x^8 - 2*x^9 - x^10 + ...
%e A230205 G.f. = q^9 - q^25 - 2*q^41 + 2*q^73 + 2*q^89 + q^121 - 2*q^137 - 2*q^153 + ...
%t A230205 a[ n_]:= SeriesCoefficient[EllipticTheta[4,0,q]*QPochhammer[-q^1,q^8]* QPochhammer[-q^7,q^8]*QPochhammer[q^8], {q, 0, n}];
%o A230205 (PARI) {a(n) = local(m, j); if( n<0, 0, m = 16*n + 9; sum( k=0, sqrtint(m \ 4), if( issquare(m - 16*k^2, &j), if( k==0, 1, 2) * (-1)^k * ((j%8)==3 || (j%8==5)))))}
%Y A230205 Cf. A030204.
%K A230205 sign
%O A230205 0,3
%A A230205 _Michael Somos_, Oct 11 2013