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A204531 Expansion of phi(q) * phi(-q^4) in powers of q where phi() is a Ramanujan theta function.

%I A204531 #17 Feb 16 2025 08:33:16
%S A204531 1,2,0,0,0,-4,0,0,-4,2,0,0,0,-4,0,0,4,4,0,0,0,0,0,0,0,6,0,0,0,-4,0,0,
%T A204531 4,0,0,0,0,-4,0,0,-8,4,0,0,0,-4,0,0,0,2,0,0,0,-4,0,0,0,0,0,0,0,-4,0,0,
%U A204531 4,8,0,0,0,0,0,0,-4,4,0,0,0,0,0,0,8,2,0,0,0,-8,0,0,0,4,0,0,0,0,0,0,0,4,0
%N A204531 Expansion of phi(q) * phi(-q^4) in powers of q where phi() is a Ramanujan theta function.
%C A204531 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A204531 G. C. Greubel, <a href="/A204531/b204531.txt">Table of n, a(n) for n = 0..1000</a>
%H A204531 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A204531 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A204531 Expansion of eta(q^2)^5 / (eta(q)^2 * eta(q^8)) in powers of q.
%F A204531 Euler transform of period 8 sequence [ 2, -3, 2, -3, 2, -3, 2, -2, ...].
%F A204531 G.f.: Product_{k>0} (1 - x^(2*k))^5 / ((1 - x^k)^2 * (1 - x^(8*k))).
%F A204531 a(4*n + 2) = a(4*n + 3) = a(8*n + 4) = 0. a(8*n) = A104794(n). a(4*n + 1) = 2 * A134343(n).
%F A204531 a(n) = (-1)^n * A246950(n). a(8*n + 1) = 2 * A113407(n). a(8*n + 5) = -4 * A053692(n). - _Michael Somos_, Jun 10 2015
%e A204531 G.f. = 1 + 2*q - 4*q^5 - 4*q^8 + 2*q^9 - 4*q^13 + 4*q^16 + 4*q^17 + ...
%t A204531 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 4, 0, q^4], {q, 0, n}]; (* _Michael Somos_, Jun 10 2015 *)
%o A204531 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 / (eta(x + A)^2 * eta(x^8 + A)), n))};
%o A204531 (PARI) {a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); 2 * (-1)^(n%8==5) * prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 2 * (e>2) * (-1)^(e<4), p%4==1, e+1, !(e%2))))};
%Y A204531 Cf. A053692, A104794, A113407, A134343, A246950.
%K A204531 sign
%O A204531 0,2
%A A204531 _Michael Somos_, Jan 15 2012