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A242609 Expansion of phi(-q) * phi(q^8) in powers of q where phi() is a Ramanujan theta function.

%I A242609 #13 Feb 16 2025 08:33:22
%S A242609 1,-2,0,0,2,0,0,0,2,-6,0,0,4,0,0,0,2,-4,0,0,0,0,0,0,4,-2,0,0,0,0,0,0,
%T A242609 2,-8,0,0,6,0,0,0,0,-4,0,0,4,0,0,0,4,-2,0,0,0,0,0,0,0,-8,0,0,0,0,0,0,
%U A242609 2,0,0,0,4,0,0,0,6,-4,0,0,4,0,0,0,0,-10,0
%N A242609 Expansion of phi(-q) * phi(q^8) in powers of q where phi() is a Ramanujan theta function.
%C A242609 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A242609 G. C. Greubel, <a href="/A242609/b242609.txt">Table of n, a(n) for n = 0..2500</a>
%H A242609 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A242609 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A242609 Expansion of eta(q)^2 * eta(q^16)^5 / (eta(q^2) * eta(q^8)^2 * eta(q^32)^2) in powers of q.
%F A242609 G.f.: (Sum_{k in Z} (-x)^k^2) * (Sum_{k in Z} (x^8)^k^2).
%F A242609 a(4*n + 2) = a(4*n + 3) = a(8*n + 5) = 0.  a(4*n) = a(8*n) = A033715(n). a(8*n + 1) = -2 * A112603(n). a(8*n + 4) = 2 * A113411(n).
%F A242609 a(n) = (-1)^n * A226225(n).
%e A242609 G.f. = 1 - 2*q + 2*q^4 + 2*q^8 - 6*q^9 + 4*q^12 + 2*q^16 - 4*q^17 + ...
%t A242609 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0, q^8], {q, 0, n}];
%o A242609 (PARI) {a(n) = if( n<1, n==0, 2 * (-1)^n * (n%4 < 2) * sumdiv( n, d, kronecker( -2, d)))};
%o A242609 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^16 + A)^5 / (eta(x^2 + A) * eta(x^8 + A)^2 * eta(x^32 + A)^2), n))};
%Y A242609 Cf. A033715, A112603, A113411, A226225.
%K A242609 sign
%O A242609 0,2
%A A242609 _Michael Somos_, May 19 2014