A256626 - OEIS

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

%S 1,1,0,0,-1,0,1,1,0,-2,-1,0,2,2,0,-2,-3,0,3,3,0,-4,-4,0,5,6,0,-6,-7,0,

%T 7,8,0,-10,-10,0,13,13,0,-14,-16,0,17,18,0,-22,-22,0,26,28,0,-30,-33,

%U 0,36,38,0,-44,-45,0,52,55,0,-60,-65,0,70,74,0,-84

%N Expansion of psi(x) / psi(x^3) in powers of x where psi() is a Ramanujan theta function.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%H G. C. Greubel, <a href="/A256626/b256626.txt">Table of n, a(n) for n = 0..1000</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 f(-x^2, -x^4) / f(-x, -x^5) in powers of x where f() is Ramanujan's two-variable theta function.

%F Expansion of q^(1/4) * eta(q^2)^2 * eta(q^3) / (eta(q) * eta(q^6)^2) in powers of

%F Euler transform of period 6 sequence [ 1, -1, 0, -1, 1, 0, ...].

%F Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = 4*v^2 - (v^2 - 1) * (u^4 - v^2).

%F Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u*v - 1)^3 - (v^4 - 1).

%F Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u^2 * (v^2 + w^2) - v*w * (3 + v^2).

%F Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = 3*u2 + u1*u3*u6 - u2^2*u6 - u1*u2*u3.

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 3^(1/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A098151.

%F G.f.: Product_{k>0} (1 + x^k + x^(2*k))^-1 * (1 - x^k + x^(2*k))^-2.

%F Convolution inverse is A101195. Convolution square is A058487. Convolution 4th power is A128633.

%F a(n) = (-1)^n * A135211(n). a(3*n + 2) = 0.

%e G.f. = 1 + x - x^4 + x^6 + x^7 - 2*x^9 - x^10 + 2*x^12 + 2*x^13 - 2*x^15 + ...

%e G.f. = 1/q + q^3 - q^15 + q^23 + q^27 - 2*q^35 - q^39 + 2*q^47 + 2*q^51 + ...

%t a[ n_] := SeriesCoefficient[ q^(1/4) EllipticTheta[ 2, 0, q^(1/2)] / EllipticTheta[ 2, 0, q^(3/2)], {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 + A) * eta(x^6 + A)^2), n))};

%Y Cf. A058487, A098151, A101195, A128633, A135211.

%K sign

%O 0,10

%A _Michael Somos_, Apr 05 2015