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A257399 Expansion of phi(x^3) * phi(-x^12) / chi(-x^4) in powers of x where phi(), chi() are Ramanujan theta functions.

%I A257399 #11 Feb 16 2025 08:33:25
%S A257399 1,0,0,2,1,0,0,2,1,0,0,2,2,0,0,0,2,0,0,0,3,0,0,2,0,0,0,2,1,0,0,4,2,0,
%T A257399 0,2,0,0,0,0,2,0,0,0,0,0,0,2,3,0,0,2,2,0,0,2,2,0,0,0,3,0,0,2,0,0,0,0,
%U A257399 2,0,0,0,2,0,0,4,2,0,0,2,0,0,0,2,0,0,0
%N A257399 Expansion of phi(x^3) * phi(-x^12) / chi(-x^4) in powers of x where phi(), chi() are Ramanujan theta functions.
%C A257399 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A257399 G. C. Greubel, <a href="/A257399/b257399.txt">Table of n, a(n) for n = 0..1000</a>
%H A257399 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A257399 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A257399 Expansion of (phi(-x^24)^2 + 2 * x^3 * psi(-x^12)^2) / chi(-x^4) in powers of x where phi(), psi() are Ramanujan theta functions.
%F A257399 Expansion of q^(-1/6) * eta(q^6)^5 * eta(q^8) / (eta(q^4) * eta(q^3)^2 * eta(q^24)) in powers of q.
%F A257399 Euler transform of period 24 sequence [ 0, 0, 2, 1, 0, -3, 0, 0, 2, 0, 0, -2, 0, 0, 2, 0, 0, -3, 0, 1, 2, 0, 0, -2, ...].
%F A257399 G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 8^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A257400.
%F A257399 a(4*n + 1) = a(4*n + 2) = 0. a(4*n) = A257398(n). a(4*n + 3) = 2 * A255317(n).
%e A257399 G.f. = 1 + 2*x^3 + x^4 + 2*x^7 + x^8 + 2*x^11 + 2*x^12 + 2*x^16 + 3*x^20 + ...
%e A257399 G.f. = q + 2*q^19 + q^25 + 2*q^43 + q^49 + 2*q^67 + 2*q^73 + 2*q^97 + ...
%t A257399 a[ n_] := SeriesCoefficient[ QPochhammer[ -x^4, x^4] EllipticTheta[ 3, 0, x^3] EllipticTheta[ 4, 0, x^12], {x, 0, n}];
%o A257399 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^5 * eta(x^8 + A) / (eta(x^4 + A) * eta(x^3 + A)^2 * eta(x^24 + A)), n))};
%Y A257399 Cf. A204531, A255317, A257398, A257400.
%K A257399 nonn
%O A257399 0,4
%A A257399 _Michael Somos_, Apr 21 2015