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A214316 Expansion of psi(x)^2 - 5 * x * psi(x^5)^2 in powers of x where psi() is a Ramanujan theta function.

%I A214316 #12 Feb 16 2025 08:33:18
%S A214316 1,-3,1,2,2,0,-7,2,0,2,2,-3,1,2,0,2,-6,0,2,0,1,-6,2,0,2,2,0,2,2,2,1,
%T A214316 -11,0,0,2,0,-6,2,2,2,0,0,3,2,0,2,-6,0,2,2,0,-6,0,0,0,4,-7,2,2,0,2,-3,
%U A214316 0,0,2,2,-6,2,0,2,2,0,3,2,0,0,-6,0,2,2,0
%N A214316 Expansion of psi(x)^2 - 5 * x * psi(x^5)^2 in powers of x where psi() is a Ramanujan theta function.
%C A214316 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A214316 S. Cooper, On Ramanujan's function k(q)=r(q)r^2(q^2), Ramanujan J., 20 (2009), 311-328; see p. 314 eq. (2.7)
%H A214316 G. C. Greubel, <a href="/A214316/b214316.txt">Table of n, a(n) for n = 0..2500</a>
%H A214316 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A214316 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A214316 Expansion of f(-x) * phi(-x) / chi(-x^5) in powers of x where phi(), chi(), f() are Ramanujan theta functions.
%F A214316 Expansion of q^(-1/4) * eta(q)^3 * eta(q^10) / (eta(q^2) * eta(q^5)) in powers of q.
%F A214316 Euler transform of period 10 sequence [ -3, -2, -3, -2, -2, -2, -3, -2, -3, -2, ...].
%F A214316 G.f. is a period 1 Fourier series which satisfies f(-1 / (40 t)) = 10 (t/i) g(t) where q = exp(2 Pi i t) and g(t) is g.f. for A094247.
%F A214316 a(9*n + 5) = a(9*n + 8) = 0. a(9*n + 2) = a(n).
%e A214316 G.f. = 1 - 3*x + x^2 + 2*x^3 + 2*x^4 - 7*x^6 + 2*x^7 + 2*x^9 + 2*x^10 - 3*x^11 + ...
%e A214316 G.f. = q - 3*q^5 + q^9 + 2*q^13 + 2*q^17 - 7*q^25 + 2*q^29 + 2*q^37 + 2*q^41 + ...
%t A214316 a[ n_] := SeriesCoefficient[ QPochhammer[ x]^3 QPochhammer[ x^10] / (QPochhammer[ x^2] QPochhammer[ x^5]), {x, 0, n}]
%t A214316 a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q^2]^2 - 5 EllipticTheta[ 2, 0, q^10]^2) / 4, {q, 0, 4 n + 1}]
%o A214316 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^10 + A) / (eta(x^2 + A) * eta(x^5 + A)), n))}
%Y A214316 Cf. A094247.
%K A214316 sign
%O A214316 0,2
%A A214316 _Michael Somos_, Jul 12 2012