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A246816 Expansion of phi(q) * phi(-q^2) * phi(-q^4) in powers of q where phi() is a Ramanujan theta function.

%I A246816 #15 Feb 16 2025 08:33:23
%S A246816 1,2,-2,-4,0,-4,0,8,-2,6,8,-4,0,-12,0,8,-4,8,-10,-12,0,-8,0,8,8,14,8,
%T A246816 -16,0,-4,0,16,6,16,-16,-8,0,-20,0,8,-8,8,16,-20,0,-20,0,16,-8,18,-10,
%U A246816 -8,0,-12,0,24,0,16,24,-12,0,-20,0,24,12,8,-16,-28,0
%N A246816 Expansion of phi(q) * phi(-q^2) * phi(-q^4) in powers of q where phi() is a Ramanujan theta function.
%C A246816 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A246816 G. C. Greubel, <a href="/A246816/b246816.txt">Table of n, a(n) for n = 0..1000</a>
%H A246816 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A246816 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A246816 Expansion of eta(q^2)^7 / (eta(q)^2 * eta(q^4) * eta(q^8)) in powers of q.
%F A246816 Euler transform of period 8 sequence [ 2, -5, 2, -4, 2, -5, 2, -3, ...].
%F A246816 a(n) = (-1)^floor(n/2) * A127786(n). a(2*n) = A246814(n). a(2*n + 1) = 2 * A246815(n).
%e A246816 G.f. = 1 + 2*q - 2*q^2 - 4*q^3 - 4*q^5 + 8*q^7 - 2*q^8 + 6*q^9 + 8*q^10 + ...
%t A246816 a[n_]:= SeriesCoefficient[EllipticTheta[3, 0, q]* EllipticTheta[3, 0, -q^2]*EllipticTheta[3, 0, -q^4], {q, 0, n}]; Table[a[n], {n, 0, 50}] (* _G. C. Greubel_, Nov 30 2017 *)
%o A246816 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^7 / (eta(x + A)^2 * eta(x^4 + A) * eta(x^8 + A)), n))};
%Y A246816 Cf. A127786, A246814, A246815.
%K A246816 sign
%O A246816 0,2
%A A246816 _Michael Somos_, Sep 03 2014