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A143377 Expansion of q^(-1/6) * eta(q)^2 * eta(q^4) / eta(q^2) in powers of q.

%I A143377 #17 Feb 16 2025 08:33:08
%S A143377 1,-2,0,0,1,2,0,0,-3,0,0,0,-2,2,0,0,2,2,0,0,-1,-2,0,0,0,-2,0,0,1,-2,0,
%T A143377 0,2,2,0,0,4,-2,0,0,-2,0,0,0,0,-2,0,0,-1,0,0,0,-2,0,0,0,2,4,0,0,-1,2,
%U A143377 0,0,0,0,0,0,-2,0,0,0,-2,2,0,0,-2,-2,0,0,0,-2,0,0,0,4,0,0,1,0,0,0,4,0,0,0,-2,0,0,0,2,-2,0,0,1
%N A143377 Expansion of q^(-1/6) * eta(q)^2 * eta(q^4) / eta(q^2) in powers of q.
%C A143377 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A143377 G. C. Greubel, <a href="/A143377/b143377.txt">Table of n, a(n) for n = 0..1000</a>
%H A143377 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A143377 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A143377 Expansion of phi(-x) * f(-x^4) = psi(-x) * f(-x) = psi(-x)^2 * chi(-x^2) = f(-x)^2 / chi(-x^2) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.
%F A143377 Euler transform of period 4 sequence [ -2, -1, -2, -2, ...].
%F A143377 a(n) = (-1)^(n / 2) * b(6*n + 1) where b() is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 8) or p == 23 (mod 24), b(p^e) = (-1)^(e/2) * (1 + (-1)^e) / 2 if p == 3 (mod 8) or p == 17 (mod 24) and p>3, b(p^e) = (e+1) * s^e if p == 1, 7 (mod 24) where p = x^2 + 6*y^2 and s = Kronecker(12, x) * (-1)^((p-1) / 12).
%F A143377 G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 4608^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A143379.
%F A143377 a(4*n + 2) = a(4*n + 3) = 0.
%F A143377 G.f.: Product_{k>0} (1 - x^k)^2 * (1 + x^(2*k)).
%F A143377 a(n) = (-1)^n * A143380(n). a(4*n) = A143378(n). a(4*n + 1) = -2 * A143379(n). - _Michael Somos_, Apr 07 2015
%e A143377 G.f. = 1 - 2*x + x^4 + 2*x^5 - 3*x68 - 2*x^12 + 2*x^13 + 2*x^16 + 2*x^17 + ...
%e A143377 G.f. = q - 2*q^7 + q^25 + 2*q^31 - 3*q^49 - 2*q^73 + 2*q^79 + 2*q^97 + 2*q^103 + ...
%t A143377 a[ n_] := SeriesCoefficient[ QPochhammer[ x^4] EllipticTheta[ 4, 0, x], {x, 0, n}]; (* _Michael Somos_, Apr 07 2015 *)
%o A143377 (PARI) {a(n) = my(A, p, e, x); if(n<0, 0, A = factor(6*n + 1); simplify( I^n * prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if(p<5, 0, p%8==5 || p%24==23, !(e%2), p%8==3 || p%24==17, (-1)^(e\2)*!(e%2), for(i=1, sqrtint(p\6), if( issquare(p - 6*i^2, &x), break)); (e+1) * (kronecker(12, x) * I^((p-1) / 6))^e ))))};
%o A143377 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A) / eta(x^2 + A), n))};
%Y A143377 Cf. A143378, A143379, A143380.
%K A143377 sign
%O A143377 0,2
%A A143377 _Michael Somos_, Aug 10 2008