cp's OEIS Frontend

A138527 Expansion of phi(-q) / phi(-q^5) in powers of q where phi() is a Ramanujan theta function.

%I A138527 #17 Feb 16 2025 08:33:08
%S A138527 1,-2,0,0,2,2,-4,0,0,2,4,-8,0,0,4,8,-14,0,0,8,14,-24,0,0,12,22,-40,0,
%T A138527 0,20,36,-64,0,0,32,56,-98,0,0,48,84,-148,0,0,72,126,-220,0,0,106,184,
%U A138527 -320,0,0,152,264,-460,0,0,216,376,-652,0,0,306,528,-912,0,0,424,732,-1264,0,0,584,1008
%N A138527 Expansion of phi(-q) / phi(-q^5) in powers of q where phi() is a Ramanujan theta function.
%C A138527 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A138527 Denoted by t in Andrews and Berndt 2005. - _Michael Somos_, Apr 25 2016
%D A138527 G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part I, Springer, New York, 2005, MR2135178 (2005m:11001) See p. 337.
%H A138527 G. C. Greubel, <a href="/A138527/b138527.txt">Table of n, a(n) for n = 0..1000</a>
%H A138527 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A138527 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A138527 Expansion of (eta(q) / eta(q^5))^2 * eta(q^10) / eta(q^2) in powers of q.
%F A138527 Euler transform of period 10 sequence [ -2, -1, -2, -1, 0, -1, -2, -1, -2, 0, ...].
%F A138527 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (v^2 - u^2)^2 - u^2 * (1 - v^2) * (5 - v^2).
%F A138527 G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (v^2 - u^2) * (u + v)^2 - u * v * (1 - u^2) * (5 - v^2).
%F A138527 G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u + v)^2 * w^2 - u * v * (5 - v^2).
%F A138527 G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = (u2 * u3 - u1 * u6)^2 - u1 * u3 * (u6^2 - u2^2).
%F A138527 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 A116494.
%F A138527 G.f.: Product_{k>0} P(10, x^k) / P(5, x^k) where P(n, x) is the n-th cyclotomic polynomial.
%F A138527 a(5*n + 2) = a(5*n + 3) = 0.
%F A138527 Convolution inverse is A138526. Convolution square is A138518.
%e A138527 G.f. = 1 - 2*q + 2*q^4 + 2*q^5 - 4*q^6 + 2*q^9 + 4*q^10 - 8*q^11 + 4*q^14 + ...
%t A138527 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] / EllipticTheta[ 4, 0, q^5], {q, 0, n}]; (* _Michael Somos_, Sep 13 2015 *)
%o A138527 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^5 + A))^2 * eta(x^10 + A) / eta(x^2 + A), n))};
%Y A138527 Cf. A116494, A138158, A138526.
%K A138527 sign
%O A138527 0,2
%A A138527 _Michael Somos_, Mar 23 2008