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A246814 Expansion of phi(-q) * phi(-q^4)^2 in powers of q where phi() is a Ramanujan theta function.

%I A246814 #11 Feb 16 2025 08:33:23
%S A246814 1,-2,0,0,-2,8,0,0,-4,-10,0,0,8,8,0,0,6,-16,0,0,-8,16,0,0,-8,-10,0,0,
%T A246814 0,24,0,0,12,-16,0,0,-10,8,0,0,-8,-32,0,0,24,24,0,0,8,-18,0,0,-8,24,0,
%U A246814 0,-16,-16,0,0,0,24,0,0,6,-32,0,0,-16,32,0,0,-12
%N A246814 Expansion of phi(-q) * phi(-q^4)^2 in powers of q where phi() is a Ramanujan theta function.
%C A246814 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A246814 G. C. Greubel, <a href="/A246814/b246814.txt">Table of n, a(n) for n = 0..1000</a>
%H A246814 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A246814 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A246814 Expansion of eta(q)^2 * eta(q^4)^4 / (eta(q^2) * eta(q^8)^2) in powers of q.
%F A246814 Euler transform of period 8 sequence [ -2, -1, -2, -5, -2, -1, -2, -3, ...].
%F A246814 a(n) = (-1)^(mod(n,4) = 1) * A116597(n).
%F A246814 a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = A212885(n). a(4*n + 1) = -(-1)^n * A005876(n).
%e A246814 G.f. = 1 - 2*q - 2*q^4 + 8*q^5 - 4*q^8 - 10*q^9 + 8*q^12 + 8*q^13 + ...
%t A246814 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^4]^2, {q, 0, n}]; Table[a[n], {n, 0, 80}]
%o A246814 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^4 / (eta(x^2 + A) * eta(x^8 + A)^2), n))};
%Y A246814 Cf. A005876, A116597, A212885.
%K A246814 sign
%O A246814 0,2
%A A246814 _Michael Somos_, Sep 03 2014