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A145708 Expansion of psi(-q) / psi(-q^5) in powers of q where psi() is a Ramanujan theta function.

%I A145708 #16 Feb 16 2025 08:33:09
%S A145708 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,
%T A145708 0,6,-2,0,-4,0,8,-3,0,-6,0,11,-5,0,-8,0,14,-6,0,-10,0,18,-6,0,-12,0,
%U A145708 22,-9,0,-16,0,28,-13,0,-21,0,36,-14,0,-25,0,44,-16,0
%N A145708 Expansion of psi(-q) / psi(-q^5) in powers of q where psi() is a Ramanujan theta function.
%C A145708 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A145708 G. C. Greubel, <a href="/A145708/b145708.txt">Table of n, a(n) for n = 0..1000</a>
%H A145708 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A145708 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A145708 Expansion of q^(1/2) * eta(q) * eta(q^4) * eta(q^10) / (eta(q^2) * eta(q^5) * eta(q^20)) in powers of q.
%F A145708 Euler transform of period 20 sequence [ -1, 0, -1, -1, 0, 0, -1, -1, -1, 0, -1, -1, -1, 0, 0, -1, -1, 0, -1, 0, ...].
%F A145708 G.f. is a period 1 Fourier series which satisfies f(-1 / (80 t)) = 5^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A036026.
%F A145708 a(5*n + 2) = a(5*n + 4) = 0.
%F A145708 G.f.: (Product_{k>0} P(5, x^k) * P(20, x^k))^(-1) where P(n, x) is the n-th cyclotomic polynomial.
%F A145708 a(n) = (-1)^n * A138532(n). a(5*n + 3) = - A036026(n).
%F A145708 Convolution square is A145740. Convolution inverse is A036026.
%F A145708 a(n) = A145723(2*n - 1). a(2*n) = A146164(n). a(2*n + 1) = - A147699(n). - _Michael Somos_, Sep 06 2015
%e A145708 G.f. = 1 - x - x^3 + x^5 - x^8 + 2*x^10 - x^13 + 2*x^15 - x^16 - 2*x^18 + ...
%e A145708 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 A145708 a[ n_] := SeriesCoefficient[ x^(1/2) EllipticTheta[ 2, Pi/4, x^(1/2)] / EllipticTheta[ 2, Pi/4, x^(5/2)], {x, 0, n}]; (* _Michael Somos_, Sep 06 2015 *)
%o A145708 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^10 + A) / (eta(x^2 + A) * eta(x^5 + A) * eta(x^20 + A)), n))};
%Y A145708 Cf. A036026, A138532, A145723, A145740, A146164.
%K A145708 sign
%O A145708 0,11
%A A145708 _Michael Somos_, Oct 17 2008