cp's OEIS Frontend

A138532 Expansion of psi(x) / psi(x^5) in powers of x where psi() is a Ramanujan theta function.

%I A138532 #18 Feb 16 2025 08:33:08
%S A138532 1,1,0,1,0,-1,0,0,-1,0,2,0,0,1,0,-2,-1,0,-2,0,3,2,0,3,0,-5,-2,0,-3,0,
%T A138532 6,2,0,4,0,-8,-3,0,-6,0,11,5,0,8,0,-14,-6,0,-10,0,18,6,0,12,0,-22,-9,
%U A138532 0,-16,0,28,13,0,21,0,-36,-14,0,-25,0,44,16,0,30,0
%N A138532 Expansion of psi(x) / psi(x^5) in powers of x where psi() is a Ramanujan theta function.
%C A138532 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A138532 B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 233, Entry 66.
%H A138532 G. C. Greubel, <a href="/A138532/b138532.txt">Table of n, a(n) for n = 0..1000</a>
%H A138532 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A138532 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A138532 Expansion of q^(1/2) * (eta(q^2) / eta(q^10))^2 * eta(q^5) / eta(q) in powers of q.
%F A138532 Euler transform of period 10 sequence [ 1, -1, 1, -1, 0, -1, 1, -1, 1, 0, ...].
%F A138532 Given g.f. A(x), then B(q) = A(q^2) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (v^2 - u^2)^2 - (u^2 - 1) * (u^2 - 5) * v^2.
%F A138532 Given g.f. A(x), then B(q) = A(q^2) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (v^2 - u^2) * (u + v)^2 - u * v * (u^2 - 1) * (v^2 - 5).
%F A138532 Given g.f. A(x), then B(q) = A(q^2) / q satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u^2 * w * (v^2 - 1) - v * (v + w)^2.
%F A138532 Given g.f. A(x), then B(q) = A(q^2) / q satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = (u1 * u6 - u2 * u3)^2 - u2 * u6 * (u3^2 - u1^2).
%F A138532 G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = 5^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A138526.
%F A138532 G.f.: (Product_{k>0} P(5,x^k) * P(10,x^k)^2)^(-1) where P(n,x) is the n-th cyclotomic polynomial.
%F A138532 a(5*n + 2) = a(5*n + 4) = 0.
%F A138532 Convolution square is A138516. Convolution inverse is A116494.
%e A138532 G.f. = 1 + x + x^3 - x^5 - x^8 + 2*x^10 + x^13 - 2*x^15 - x^16 - 2*x^18 + ...
%e A138532 G.f. = 1/q + q + q^5 - q^9 - q^15 + 2*q^19 + q^25 - 2*q^29 - q^31 - 2*q^35 + ...
%t A138532 a[ n_] := SeriesCoefficient[ x^(1/2) EllipticTheta[ 2, 0, x^(1/2)] / EllipticTheta[ 2, 0, x^(5/2)], {x, 0, n}]; (* _Michael Somos_, Sep 08 2015 *)
%o A138532 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^5 + A) / eta(x + A) * (eta(x^2 + A) / eta(x^10 + A))^2, n))};
%Y A138532 Cf. A116494, A138516.
%K A138532 sign
%O A138532 0,11
%A A138532 _Michael Somos_, Mar 23 2008