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%I #49 Mar 12 2021 22:24:47
%S 1,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,1,0,0,2,0,0,0,0,0,0,0,2,2,0,0,0,1,0,
%T 0,2,0,0,0,0,0,0,0,2,2,0,0,0,2,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,2,0,0,0,
%U 0,0,0,0,0,0,0,2,0,0,0,0,3,0,0,2,0,0,0
%N Expansion of phi(x^3) * phi(-x^48) / chi(-x^16) in powers of x where phi(), chi() 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="/A256505/b256505.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^(-2/3) * eta(q^6)^5 * eta(q^32) * eta(q^48)^2 / (eta(q^3)^2 * eta(q^12)^2 * eta(q^16) * eta(q^96)) in powers of q.
%F Euler transform of a period 96 sequence.
%F a(n) = A257403(3*n + 2) unless n == 5 (mod 8).
%F a(4*n + 1) = a(4*n + 2) = a(8*n + 7) = a(16*n + 4) = a(16*n + 8) = 0.
%F a(4*N) = A257399(n). a(8*n+3) = 2*A255318(n). a(16*n) = A257398(n). a(16*n+12) = 2*A255317(n).
%e G.f. = 1 + 2*x^3 + 2*x^12 + x^16 + 2*x^19 + 2*x^27 + 2*x^28 + x^32 + ...
%e G.f. = q^2 + 2*q^11 + 2*q^38 + q^50 + 2*q^59 + 2*q^83 + 2*q^86 + q^98 + ...
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^3] EllipticTheta[ 4, 0, x^48] QPochhammer[ -x^16, x^16], {x, 0, n}];
%o (PARI) {a(n) = my(A, p, e); if( n<0 || n%8 == 5, 0, A = factor(3*n + 2); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, p+e==3, p%8 > 4, 1-e%2, e+1)))};
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^5 * eta(x^32 + A) * eta(x^48 + A)^2 / (eta(x^3 + A)^2 * eta(x^12 + A)^2 * eta(x^16 + A) * eta(x^96 + A)), n))};
%Y Cf. A255317, A255318, A257398, A257399, A257403.
%K nonn
%O 0,4
%A _Michael Somos_, Apr 22 2015
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