%I #13 Mar 12 2021 22:24:47

%S 1,0,-2,-1,0,2,0,0,2,-1,0,0,0,0,0,0,0,-2,-1,0,-2,2,0,0,0,0,2,2,0,0,1,

%T 0,0,0,0,-2,-2,0,2,0,0,-2,0,0,0,-1,0,2,-2,0,0,0,0,0,0,0,0,0,0,2,0,0,2,

%U 1,0,2,0,0,-2,0,0,-2,2,0,0,-2,0,-2,0,0,-2

%N Expansion of psi(-x^3) * phi(-x^2) in powers of x where phi(), psi() 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="/A246962/b246962.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^(-3/8) * eta(q) * eta(q^4) * eta(q^6)^2 / (eta(q^2) * eta(q^12)) in powers of q.

%F Euler transform of period 12 sequence [0, -2, -1, -1, 0, -2, 0, -1, -1, -2, 0, -2, ...].

%F a(3*n) = A226860(n). a(3*n + 1) = 0.

%F a(3*n + 2) = -2 * A257469(n). - _Michael Somos_, Apr 25 2015

%e G.f. = 1 - 2*x^2 - x^3 + 2*x^5 + 2*x^8 - x^9 - 2*x^17 - x^18 - 2*x^20 + ...

%e G.f. = q^3 - 2*q^19 - q^27 + 2*q^43 + 2*q^67 - q^75 - 2*q^139 - q^147 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^2] EllipticTheta[ 2, Pi/4, q^(3/2)] / (Sqrt[2] q^(3/8)), {q, 0, n}];

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^12 + A) / (eta(x^4 + A) * eta(x^6 + A)), n))};

%o (PARI) {a(n) = my(A, p, e, i); if( n<0, 0, n = 8*n + 3; A = factor(n); -I * prod( k=1, matsize(A)[1], if( p = A[k,1], e = A[k,2]; if( p==2, 0, if( p==3, I^e, if( p%24 == 1 || p%24==19, for(j=1, sqrtint(p\18), if( issquare( p - 18*j^2, &i), break)); (e+1) * (if(p%24==1, 1, -I) * kronecker( 12, i))^e, if( e%2, 0, if(p%24>12, 1, -1)^(e/2)))) ))))};

%Y Cf. A226860, A257469.

%K sign

%O 0,3

%A _Michael Somos_, Sep 08 2014