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A226225 Expansion of phi(q) * phi(q^8) in powers of q where phi() is a Ramanujan theta function.

%I A226225 #21 Feb 16 2025 08:33:19
%S A226225 1,2,0,0,2,0,0,0,2,6,0,0,4,0,0,0,2,4,0,0,0,0,0,0,4,2,0,0,0,0,0,0,2,8,
%T A226225 0,0,6,0,0,0,0,4,0,0,4,0,0,0,4,2,0,0,0,0,0,0,0,8,0,0,0,0,0,0,2,0,0,0,
%U A226225 4,0,0,0,6,4,0,0,4,0,0,0,0,10,0,0,0,0
%N A226225 Expansion of phi(q) * phi(q^8) in powers of q where phi() is a Ramanujan theta function.
%C A226225 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A226225 Seiichi Manyama, <a href="/A226225/b226225.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..2500 from G. C. Greubel)
%H A226225 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A226225 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A226225 Expansion of (eta(q^2) * eta(q^16))^5 / (eta(q) * eta(q^4) * eta(q^8) * eta(q^32))^2 in powers of q.
%F A226225 Euler transform of period 32 sequence [2, -3, 2, -1, 2, -3, 2, 1, 2, -3, 2, -1, 2, -3, 2, -4, 2, -3, 2, -1, 2, -3, 2, 1, 2, -3, 2, -1, 2, -3, 2, -2, ...].
%F A226225 G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 32^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
%F A226225 G.f.: (Sum_{k in Z} x^k^2) * (Sum_{k in Z} (x^8)^k^2).
%F A226225 a(4*n + 2) = a(4*n + 3) = a(8*n + 5) = 0.  a(4*n) = a(8*n) = A033715(n). a(4*n + 1) = A033715(4*n + 1). a(8*n + 1) = 2 * A112603(n). a(8*n + 4) = 2 * A113411(n).
%F A226225 (-1)^n * a(n) = A242609(n). - _Michael Somos_, Feb 20 2015
%e A226225 G.f. = 1 + 2*q + 2*q^4 + 2*q^8 + 6*q^9 + 4*q^12 + 2*q^16 + 4*q^17 + 4*q^24 + 2*q^25 + ...
%t A226225 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^8], {q, 0, n}];
%o A226225 (PARI) {a(n) = if( n<1, n==0, 2 * (n%4 < 2) * sumdiv( n, d, kronecker( -2, d)))};
%o A226225 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^16 + A))^5 / (eta(x + A) * eta(x^4 + A) * eta(x^8 + A) * eta(x^32 + A))^2, n))};
%Y A226225 Cf. A033715, A112603, A113411, A242609.
%K A226225 nonn
%O A226225 0,2
%A A226225 _Michael Somos_, May 31 2013