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%I #48 Mar 12 2021 22:24:46
%S 1,-2,0,0,2,0,0,0,0,-2,0,0,-1,2,0,0,0,0,0,0,0,2,0,0,-1,0,0,0,-4,0,0,0,
%T 0,2,0,0,2,2,0,0,-2,0,0,0,0,0,0,0,-2,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,4,
%U 0,0,0,0,-2,0,0,0,2,0,0,0,0,0,0,0,-2,0
%N Expansion of f(-x^12) * phi(-x) in powers of x where f(), phi() are Ramanujan theta functions.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A204010/b204010.txt">Table of n, a(n) for n = 0..2500</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of q^(-1/2) * eta(q)^2 * eta(q^12) / eta(q^2) in powers of q.
%F Euler transform of period 12 sequence [-2, -1, -2, -1, -2, -1, -2, -1, -2, -1, -2, -2, ...].
%F G.f.: (Product_{k>0} 1 - x^(12*k)) * (Sum_{k in Z} (-x)^k^2).
%F a(3*n + 2) = a(4*n + 2) = a(4*n + 3) = 0.
%e G.f. = 1 - 2*x + 2*x^4 - 2*x^9 - x^12 + 2*x^13 + 2*x^21 - x^24 - 4*x^28 + 2*x^33 + ...
%e G.f. = q - 2*q^3 + 2*q^9 - 2*q^19 - q^25 + 2*q^27 + 2*q^43 - q^49 - 4*q^57 + 2*q^67 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ q^12] EllipticTheta[ 4, 0, q], {q, 0, n}];
%t a[ n_] := Module[ {m = 2 n + 1, i}, If[ n < 0, 0, Sum[i = Sqrt[ m - 2 j^2]; If[ IntegerQ @ i, (1 + Boole[ j > 0]) (-1)^j JacobiSymbol[ 12, i], 0], {j, 0, Sqrt[ m / 2]}]]];
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^12 + A) / eta(x^2 + A), n))};
%o (PARI) {a(n) = my(m, i); if( n<0, 0, m = 2*n + 1; sum( j=0, sqrtint( m \ 2), if( issquare( m - 2 * j^2, &i), (1 + (j>0)) * (-1)^j * kronecker( 12, i), 0)))};
%o (PARI) {a(n) = my(A, p, e, i); if( n<0, 0, n = 2*n + 1; A = factor(n); [1, 2*I, 0, I, 2, 0][ n\2%6 + 1] * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0, p==3, -(-I)^e, p%24 == 1 || p%24 == 19, forstep( j = 3*(p%24>1), sqrtint( p \ 2), 6, if( issquare( p - 2*j^2, &i), break)); kronecker( 12, i)^e * if(p%24>1, I^e, 1) * (e+1), if( e%2, 0, I^((p%24<12)*e)))))};
%K sign
%O 0,2
%A _Michael Somos_, Jun 03 2013
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