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A215597 Expansion of psi(-x) * f(-x)^3 in powers of x where psi(), f() are Ramanujan theta functions.

%I A215597 #15 Feb 16 2025 08:33:18
%S A215597 1,-4,3,4,-2,0,-11,4,0,12,10,-12,-7,-4,0,-12,16,0,6,0,9,8,-10,0,-18,
%T A215597 -20,0,20,-14,12,11,24,0,0,-22,0,16,-20,-6,-12,0,0,-3,4,0,-20,48,0,14,
%U A215597 28,0,-40,0,0,0,-8,-33,-4,-26,0,30,28,0,0,2,12,-16,20,0
%N A215597 Expansion of psi(-x) * f(-x)^3 in powers of x where psi(), f() are Ramanujan theta functions.
%C A215597 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A215597 G. C. Greubel, <a href="/A215597/b215597.txt">Table of n, a(n) for n = 0..1000</a>
%H A215597 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A215597 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A215597 Expansion of q^(-1/4) * eta(q)^4 * eta(q^4) / eta(q^2) in powers of q.
%F A215597 Euler transform of period 4 sequence [ -4, -3, -4, -4, ...].
%F A215597 G.f. is a period 1 Fourier series which satisfies f(-1 / (128 t)) = 2^(19/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A215596.
%F A215597 a(n) = (-1)^floor( n/2 ) * b(4*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 * (-p)^(e/2) if p == 3 (mod 4),
%F A215597 Convolution of A106459 and A010816.
%e A215597 1 - 4*x + 3*x^2 + 4*x^3 - 2*x^4 - 11*x^6 + 4*x^7 + 12*x^9 + 10*x^10 + ...
%e A215597 q - 4*q^5 + 3*q^9 + 4*q^13 - 2*q^17 - 11*q^25 + 4*q^29 + 12*q^37 + 10*q^41 + ...
%t A215597 A215597[n_] := SeriesCoefficient[(QPochhammer[x]^4 * QPochhammer[x^4])/ QPochhammer[x^2], {x, 0, n}]; Table[A215597[n], {n, 0, 50}] (* _G. C. Greubel_, Oct 01 2017 *)
%o A215597 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^4 + A) / eta(x^2 + A), n))}
%Y A215597 Cf. A010816, A106459, A215596.
%K A215597 sign
%O A215597 0,2
%A A215597 _Michael Somos_, Aug 16 2012