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A093085 Expansion of phi(-x) / psi(x^4) in powers of x where phi(), psi() are Ramanujan theta functions.

%I A093085 #21 Feb 16 2025 08:32:53
%S A093085 1,-2,0,0,1,2,0,0,-1,-4,0,0,0,6,0,0,1,-8,0,0,0,12,0,0,-1,-18,0,0,-1,
%T A093085 24,0,0,2,-32,0,0,1,44,0,0,-2,-58,0,0,-1,76,0,0,2,-100,0,0,1,128,0,0,
%U A093085 -3,-164,0,0,-1,210,0,0,4,-264,0,0,2,332,0,0,-5,-416,0,0,-2,516,0,0,5,-640,0,0,2,790,0,0,-6,-968
%N A093085 Expansion of phi(-x) / psi(x^4) in powers of x where phi(), psi() are Ramanujan theta functions.
%C A093085 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A093085 eta(q^2) * eta(q^8)^6 = eta(q)^2 * eta(q^4)^2 * eta(q^8) * eta(q^16)^2 + 2 * eta(q^2) * eta(q^4)^2 * eta(q^16)^4 is equivalent to the a(4*n), ..., a(4*n + 3) results.
%H A093085 G. C. Greubel, <a href="/A093085/b093085.txt">Table of n, a(n) for n = 0..1000</a>
%H A093085 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A093085 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A093085 Expansion of q^(1/2) * eta(q)^2 * eta(q^4) / (eta(q^2) * eta(q^8)^2) in powers of q.
%F A093085 Euler transform of period 8 sequence [ -2, -1, -2, -2, -2, -1, -2, 0, ...].
%F A093085 Given g.f. A(x), then B(q) = A(q)^2 / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u * (u + 8) * (v + 4) - v^2.
%F A093085 G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 32^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A187154.
%F A093085 G.f.: Product_{k>0} (1 - x^k)^2 / ((1 - x^(4*k - 2)) * (1 - x^(8*k))^2).
%F A093085 a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = A029838(n). a(4*n + 1) =  -2 * A083365(n). Convolution square is A131124. Convolution inverse is A187154.
%e A093085 G.f. = 1 - 2*x + x^4 + 2*x^5 - x^8 - 4*x^9 + 6*x^13 + x^16 - 8*x^17 + 12*x^21 - ...
%e A093085 G.f. = 1/q - 2*q + q^7 + 2*q^9 - q^15 - 4*q^17 + 6*q^25 + q^31 - 8*q^33 + ...
%t A093085 a[ n_] := SeriesCoefficient[ 2 x^(1/2) EllipticTheta[ 4, 0, x] / EllipticTheta[ 2, 0, x^2], {x, 0, n}];
%o A093085 (PARI) {a(n) = if( n<0, 0, polcoeff( prod( k=1, n, (1 - x^k)^[ 0, 2, 1, 2, 2, 2, 1, 2][1 + k%8], 1 + x * O(x^n)), n))}
%o A093085 (PARI) {a(n) = my(A, A2, m); if( n<0, 0, A = x + O(x^2); m=1; while( m<=n, m*=2; A = subst(A, x, x^2); A = 4*A + 16*A^2 + (1 + 8*A) * sqrt(A + 4*A^2)); polcoeff( sqrt(x / A), n))}
%o A093085 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A) / (eta(x^2 + A) * eta(x^8 + A)^2), n))}
%Y A093085 Cf. A029838, A029839, A083365, A131124, A187154.
%K A093085 sign
%O A093085 0,2
%A A093085 _Michael Somos_, Mar 20 2004, Oct 22 2007