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A116494 Expansion of psi(q^5)/psi(q) in powers of q where psi() is a Ramanujan theta function.

%I A116494 #13 Feb 16 2025 08:33:00
%S A116494 1,-1,1,-2,3,-3,4,-6,8,-10,12,-16,21,-25,30,-38,48,-57,68,-84,102,
%T A116494 -121,143,-172,207,-243,284,-338,400,-465,542,-636,744,-862,996,-1158,
%U A116494 1344,-1546,1776,-2050,2361,-2701,3088,-3540,4050,-4613,5248,-5980,6808,-7719,8742,-9916,11232,-12682
%N A116494 Expansion of psi(q^5)/psi(q) in powers of q where psi() is a Ramanujan theta function.
%C A116494 Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
%H A116494 G. C. Greubel, <a href="/A116494/b116494.txt">Table of n, a(n) for n = 0..1000</a>
%H A116494 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A116494 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A116494 Expansion of q^(-1/2)*eta(q)*eta(q^10)^2/(eta(q^2)^2*eta(q^5)) in powers of q.
%F A116494 Euler transform of period 10 sequence [ -1,1,-1,1,0,1,-1,1,-1,0,...].
%F A116494 Given g.f. A(x), then B(x)=x*A(x^2) satisfies 0=f(B(x),B(x^2)) where f(u,v)=(1-u^2)(1-5u^2)v^2 -(u^2-v^2)^2.
%F A116494 Given g.f. A(x), then B(x)=x*A(x^2) satisfies 0=f(B(x),B(x^2),B(x^4)) where f(u,v,w)=v*w*(1-v^2)-u^2*(v+w)^2.
%F A116494 Given g.f. A(x), then B(x)=x*A(x^2) satisfies 0=f(B(x),B(x^2),B(x^3),B(x^6)) where f(u1,u2,u3,u6)=u2*u6*(u1^2-u3^2) -(u2*u3-u1*u6)^2.
%F A116494 G.f.: Product_{k>0} (1-x^k)/(1-x^(5k))*((1-x^(10k))/(1-x^(2k)))^2 = (Sum_{k>0} x^(5(k^2-k)/2))/(Sum_{k>0} x^((k^2-k)/2)).
%F A116494 a(n) = (-1)^n*A036026(n).
%t A116494 a[n_]:= SeriesCoefficient[q^(-1/2)*(EllipticTheta[2, 0, q^(5/2)]/EllipticTheta[2, 0, q^(1/2)]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* _G. C. Greubel_, Jan 04 2018 *)
%o A116494 (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)*eta(x^10+A)^2/eta(x^2+A)^2/eta(x^5+A), n))}
%Y A116494 Cf. A036026.
%K A116494 sign
%O A116494 0,4
%A A116494 _Michael Somos_, Feb 18 2006