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A246608 Expansion of phi(-q) * phi(-q^4)^4 in powers of q where phi() is a Ramanujan theta function.

%I A246608 #25 Feb 16 2025 08:33:23
%S A246608 1,-2,0,0,-6,16,0,0,8,-50,0,0,16,80,0,0,-38,-96,0,0,-16,160,0,0,48,
%T A246608 -242,0,0,64,240,0,0,-56,-288,0,0,-150,400,0,0,112,-384,0,0,112,496,0,
%U A246608 0,-112,-674,0,0,-80,560,0,0,160,-672,0,0,192,880,0,0,-294
%N A246608 Expansion of phi(-q) * phi(-q^4)^4 in powers of q where phi() is a Ramanujan theta function.
%C A246608 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A246608 G. C. Greubel, <a href="/A246608/b246608.txt">Table of n, a(n) for n = 0..1000</a>
%H A246608 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A246608 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A246608 Expansion of eta(q)^2 * eta(q^4)^8 / (eta(q^2) * eta(q^8)^4) in powers of q.
%F A246608 a(4*n) = A245643(n). a(4*n + 1) = -2 * A244276(n). a(4*n + 2) = a(4*n + 3) = 0.
%e A246608 G.f. = 1 - 2*q - 6*q^4 + 16*q^5 + 8*q^8 - 50*q^9 + 16*q^12 + 80*q^13 + ...
%t A246608 a[n_]:= SeriesCoefficient[EllipticTheta[3,0, -q]*EllipticTheta[3,0, -q^4 ]^4, {q, 0, n}]; (* corrected by _G. C. Greubel_, Mar 15 2018 *)
%o A246608 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^8 / (eta(x^2 + A) * eta(x^8 + A)^4), n))};
%o A246608 (Magma) A := Basis( ModularForms( Gamma0(8), 5/2), 68); A[1] - 2*A[2];
%Y A246608 Cf. A244276, A245643.
%K A246608 sign
%O A246608 0,2
%A A246608 _Michael Somos_, Sep 01 2014