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A143379 Expansion of q^(-7/24) * eta(q) * eta(q^4)^2 / eta(q^2) in powers of q.

%I A143379 #18 Feb 16 2025 08:33:08
%S A143379 1,-1,0,-1,-1,1,1,1,-1,1,0,1,0,0,-2,-1,0,0,-1,1,1,-2,0,0,0,1,1,0,2,0,
%T A143379 1,-1,-1,0,1,-1,0,0,1,0,-1,-1,0,-1,-1,-1,0,0,0,1,0,1,0,1,-1,-1,2,0,-1,
%U A143379 1,-1,1,0,3,1,-1,0,0,0,1,-2,0,0,-1,-1,0,-1,0,1,0,0,1,-1,-1,-1,0,0,0,0,-1,0,-2,0,1,2,1,-1,0,2,1,0,0,0,0,1
%N A143379 Expansion of q^(-7/24) * eta(q) * eta(q^4)^2 / eta(q^2) in powers of q.
%C A143379 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A143379 G. C. Greubel, <a href="/A143379/b143379.txt">Table of n, a(n) for n = 0..1000</a>
%H A143379 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A143379 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A143379 Expansion of psi(x^2) * f(-x) = psi(-x) * f(-x^4) = chi(-x) * f(-x^4)^2 = psi(-x)^2 / chi(-x) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions. - _Michael Somos_, Apr 07 2015
%F A143379 Euler transform of period 4 sequence [ -1, 0, -1, -2, ...].
%F A143379 G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 72^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A143377.
%F A143379 G.f.: Product_{k>0} (1 - x^(4*k))^2 * (1 - x^(2*k-1)).
%F A143379 Convolution of A000009 and A134343. - _Michael Somos_, Jul 11 2012
%F A143379 -2 * a(n) = A143377(4*n + 1). 2 * a(n) = A143380(4*n + 1).
%F A143379 a(2*n) = A214302(n). a(2*n + 1) = - A214303(n). - _Michael Somos_, Jul 11 2012
%e A143379 G.f. = 1 - x - x^3 - x^4 + x^5 + x^6 + x^7 - x^8 + x^9 + x^11 - 2*x^14 - x^15 - x^18 + ...
%e A143379 G.f. = q^7 - q^31 - q^79 - q^103 + q^127 + q^151 + q^175 - q^199 + q^223 + ...
%t A143379 a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^4]^2 / QPochhammer[ x^2], {x, 0, n}]; (* _Michael Somos_, Jul 11 2012 *)
%t A143379 a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] QPochhammer[ x^4]^2, {x, 0, n}]; (* _Michael Somos_, Apr 07 2015 *)
%o A143379 (PARI) {a(n) = my(A, p, e, x); if( n<0, 0, n = n*4 + 1; A = factor(6*n + 1); simplify( I^n / -2 * 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 A143379 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A), n))};
%Y A143379 Cf. A000009, A134343, A143377, A143380.
%K A143379 sign
%O A143379 0,15
%A A143379 _Michael Somos_, Aug 11 2008