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A116498 Expansion of psi(-q)/psi(-q^2) in powers of q where psi() is a Ramanujan theta function.

%I A116498 #13 Feb 16 2025 08:33:00
%S A116498 1,-1,1,-2,1,-2,3,-3,4,-5,6,-7,8,-9,11,-13,16,-18,21,-24,27,-32,36,
%T A116498 -41,48,-54,61,-70,78,-88,100,-112,127,-143,159,-179,199,-222,248,
%U A116498 -276,308,-342,380,-421,465,-516,570,-629,697,-767,845,-932,1022,-1124,1236,-1355,1488,-1631,1785,-1954,2136
%N A116498 Expansion of psi(-q)/psi(-q^2) in powers of q where psi() is a Ramanujan theta function.
%C A116498 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 A116498 G. C. Greubel, <a href="/A116498/b116498.txt">Table of n, a(n) for n = 0..1000</a>
%H A116498 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A116498 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A116498 Expansion of q^(1/8)*eta(q)*eta(q^4)^2/(eta(q^2)^2*eta(q^8)) in powers of q.
%F A116498 a(n)=(-1)^n*A070048(n).
%F A116498 Euler transform of period 8 sequence [ -1,1,-1,-1,-1,1,-1,0,...].
%F A116498 Given g.f. A(x), then B(x)=A(x^8)/x satisfies 0=f(B(x),B(x^3)) where f(u,v)=3*u*v -(u+v^3)*(v-u^3).
%F A116498 G.f.: Product_{k>0} (1+x^(2k))/((1+x^k)(1+x^(4k))) = (Sum_{k>0} (-x)^((k^2-k)/2))/(Sum_{k>0} (-x^2)^((k^2-k)/2)).
%t A116498 eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[ q^(1/8)* eta[q]*eta[q^4]^2/(eta[q^2]^2*eta[q^8]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* _G. C. Greubel_, Mar 07 2018 *)
%o A116498 (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)*eta(x^4+A)^2/eta(x^2+A)^2/eta(x^8+A), n))}
%Y A116498 Cf. A000122, A000700, A010054, A070048, A121373.
%K A116498 sign
%O A116498 0,4
%A A116498 _Michael Somos_, Feb 18 2006